A Grothendieck universe is known in set theory as the set Vκ for a (strongly) inaccessible cardinal κ. They are exactly the same thing. Thus, the existence of a Grothendieck universe is exactly equivalent to the existence of one inaccessible cardinal. These cardinals and the corresponding universes have been studied in set theory for over a century.
The Grothendieck Universe axiom (AU) is the assertion that every set is an element of a universe in this sense. Thus, it is equivalent to the assertion that the inaccessible cardinals are unbounded in the cardinals. In other words, that there is a proper class of inaccessible cardinals. This is the axiom you sought, which is exactly equivalent to AU. In this sense, the axiom AU is a statement in set theory, having nothing necessarily to do with category theory.
The large cardinal axioms are fruitfully measured in strength not just by direct implication, but also by their consistency strength. One large cardinal property LC1 is stronger than another LC2 in consistency strength if the consistency of ZFC with an LC1 large cardinal implies the consistency of ZFC with an LC2 large cardinal.
Measured in this way, the AU axiom has a stronger consistency strength than the existence of any finite or accessible number of inaccessible cardinals, and so one might think it rather strong. But actually, it is much weaker than the existence of a single Mahlo cardinal, the traditional next-step-up in the large cardinal hierarchy. The reason is that if κ is Mahlo, then κ is a limit of inaccessible cardinals, and so Vκ will satisfy ZFC plus the AU axiom. The difference between AU and Mahloness has to do with the thickness of the class of inaccessible cardinals. For example, strictly stronger than AU and weaker than a Mahlo cardinal is the assertion that the inaccessible cardinals form a stationary proper class, an assertion known as the Levy Scheme (which is provably equivconsistent with some other interesting axioms of set theory, such as the boldface Maximality Principle, which I have studied a lot). Even Mahlo cardinals are regarded as rather low in the large cardinal hierarchy, far below the weakly compact cardinals, Ramsey cardinals, measurable cardinals, strong cardinals and supercompact cardinals. In particular, if δ is any of these large cardinals, then δ is a limit of Mahlo cardinals, and certainly a limit of strongly inaccessible cardinals. So in particular, Vδ will be a model of the AU axiom.
Rather few of the large cardinal axioms imnply AU directly, since most of them remain true if one were to cut off the universe at a given inaccessible cardinal, a process that kills AU. Nevertheless, implicit beteween levels of the large caridnal hiearchy are the axioms of the same form as AU, which assert an unbounded class of the given cardinal. For example, one might want to have unboundedly many Mahlo cardinals, or unboundedly many measurable cardinals, and so on. And the consistency strength of these axioms is still below the consistency strength of a single supercompact cardinal. The hierarchy is extremely fine and intensely studied. For example, the assertion that there are unboundedly many strong cardinals is equiconsistent with the impossiblity to affect projective truth by forcing. The existence of a proper class of Woodin cardinals is particularly robust set-theoretically, and all of these axioms are far stronger than AU.
There are natural weakenings of AU that still allow for almost all if not all of what category theorists do with these universes. Namely, with the universes, it would seem to suffice for almost all category-theoretic purposes, if a given universe U were merely a model of ZFC, rather than Vκ for an inaccessible cardinal κ. The difference is that U is merely a model of the Power set axiom, rather than actually being closed under the true power sets (and similarly using Replacement in place of regularity). The weakening of AU I have in mind is the axiom that asserts that every set is an element of a transitive model of ZFC. This assertion is strictly weaker in consistency strength thatn even a single inaccessible cardinal. One can get much lower, if one weakens the concept of universe to just a fragment of ZFC. Then one could arrive at a version of AU that was actually provable in ZFC, but which could be used for most all of the applications in cateogory theory to my knowledge. In this sense, ZFC itself is a kind of large cardinal axiom relative to the weaker fragments of ZFC.
This is a nice problem. Here is what I know.
(Below, I refer to the Handbook. This is the Handbook of Set Theory, Foreman, Kanamori, eds., Springer, 2010.)
First of all, the consistency of the failure of diamond at a weakly compact cardinal seems open. Woodin has asked this explicitly, I do not know if the question itself is due to him. Of course, $\diamondsuit_\kappa$ holds if $\kappa$ is measurable, so the problem is delicate. In fact, measurability is an overkill, and it suffices that $\kappa$ is subtle. The notion of subtlety is due to Jensen and Kunen, and the fact that $\diamondsuit_\kappa$ holds for subtle cardinals is due to Kunen, see
Ronald B. Jensen, and Kenneth Kunen. Some Combinatorial Properties of $L$ and $V$, unpublished manuscript, 1969, currently available at Jensen's page.
The argument in the Jensen-Kunen paper actually shows that subtlety of $\kappa$ implies $\diamondsuit_\kappa(\mathsf{REG})$, where $\mathsf{REG}$ denotes the stationary set of regular cardinals below $\kappa$. On the other hand, Woodin showed that it is equiconsistent with the existence of a weakly compact cardinal that $\mathsf{GCH}$ holds, and there is a weakly compact $\kappa$ such that $\diamondsuit_\kappa(\mathsf{REG})$ fails.
I do not know of the precise date of this result, but it has been extended in a variety of ways. For example, for $m,n\ge 1$, we can replace weak compactness with $\Pi^m_n$-indescribability. This stronger result is due to Hauser, see
Kai Hauser. Indescribable cardinals without diamonds, Arch. Math. Logic, 31 (5), (1992), 373–383. MR1164732 (93b:03082).
Džamonja and Hamkins further extended this to strongly unfoldable cardinals, see
Mirna Džamonja, and Joel David Hamkins. Diamond (on the regulars) can fail at any strongly unfoldable cardinal, Ann. Pure Appl. Logic, 144 (1-3), (2006), 83–95. MR2279655 (2007m:03091).
(But, again, this is not $\lnot\diamondsuit_\kappa(\kappa)$.) Another extension is due to Hellsten, see
Alex Hellsten. Diamonds on large cardinals, Thesis (Ph.D.)–Helsingin Yliopisto (Finland). 2003. 72 pp.
In his thesis, Hellsten proves the consistency of the failure of what he calls weakly compact diamond. To define this principle, let $\kappa$ be weakly compact. Consider the normal ideal over $\kappa$ generated by the sets of the form $\{\alpha<\kappa\mid (V_\alpha,\in, U\cap V_\alpha)\models\lnot\phi\}$ where $U\subseteq V_\kappa$, $\phi$ is a $\Pi^1_1$-sentence, and $(V_\kappa,\in,U)\models\phi$. Call reflective those subsets of $\kappa$ that are positive with respect to this ideal. We say that weakly compact diamond holds at $\kappa$ iff $\kappa$ is weakly compact and there is a sequence $(A_\alpha\mid \alpha<\kappa)$ such that $A_\alpha\subseteq\alpha$ for all $\alpha$, and for any $A\subseteq\kappa$ we have that $\{\alpha\mid A_\alpha=A\cap\alpha\}$ is reflective. (This principle was studied independently by Sun and Shelah; Hauser's results also give the consistency of the existence of weakly compact cardinals where weakly compact diamond fails.)
On the other hand, it is consistent that diamond fails at an inaccessible, or a Mahlo, or even a greatly Mahlo cardinal.
By results of Jensen, if diamond fails at a Mahlo, then $0^\sharp$ exists, the reference is
Ronald B. Jensen. Diamond at Mahlo cardinals, handwritten notes, Oberwolfach, 1991.
(Martin Zeman may have a copy of these notes.) In the note, Jensen proves that $0^\sharp$ exists if there is a $\kappa$ Mahlo such that $\diamondsuit_\kappa(\mathrm{cof}(\omega_1))$ fails.
This indicates that the arguments must be of a different nature than the equiconsistencies mentioned above. The best current lower bound is due to Zeman, see
Martin Zeman. $\diamondsuit$ at Mahlo cardinals, J. Symbolic Logic, 65 (4), (2000), 1813–1822. MR1812181 (2002g:03107).
Martin proves that if there is a Mahlo cardinal $\kappa$, and a regular cardinal $\varepsilon$ such that $\omega_1<\varepsilon<\kappa$, and $\diamondsuit_\kappa(\mathrm{cof}(\varepsilon))$ fails, then either $0^\mathbf{P}$, zero-pistol, exists, or else (the core model $K$ exists, and) $o^K(\beta)\ge\varepsilon$ for stationarily many $\beta<\kappa$. For difficulties improving this result, see the remarks in
Martin Zeman. Diamond, GCH and weak square, Proc. Amer. Math. Soc., 138 (5), (2010), 1853–1859. MR2587470 (2011g:03114).
The consistency of the failure of diamond at an inaccessible, a Mahlo, or a greatly Mahlo cardinal is due to Woodin. I do not know how efficient the upper bounds he obtains are, they are strong versions of hypermeasurability, somewhat past the existence of a $\kappa$ with Mitchell order $o(\kappa)=\kappa^{++}$, so there is certainly room for improvement bewteen these bounds and Zeman's.
The argument uses Radin forcing. James Cummings has a nice write-up (from around 1995), that I'm sure he will give you a copy of, if you email him directly,
James Cummings. Woodin's theorem on killing diamonds via Radin forcing, unpublished notes, 1995.
The idea is to begin with an embedding whose associated Radin sequence is sufficiently large. Recall that given $j:V\to M$ elementary, with critical point $\kappa$, we can define the Radin sequence associated to $j$ by setting $u^j(0)=\kappa$, and for $\alpha>0$, $u^j(\alpha)=\{X\subseteq V_\kappa\mid u^j\upharpoonright \alpha\in j(X)\}$. The construction stops at the first $\alpha$ such that $u^j\upharpoonright\alpha\notin M\cap V_{j(\kappa)}$. For $\beta<\alpha$, these $u^j(\beta)$ are $\kappa$-complete non-principal ultrafilters on $V_\kappa$. We call $\alpha$ the length of $u$, $\mathrm{lh}(u)$.
I'll skip here the definition of the Radin forcing $\mathbb R_u$ associated to $u$, as it is somewhat technical. It is not the original version as in Radin's paper, since he assumes supercompactness. A good reference is Gitik's paper in the Handbook, or see Chapter 6 in the unpublished
James Cummings, and W. Hugh Woodin. Generalised Prikry forcing, draft of book in preparation, ca. 1995.
(This is the book I refer to in this answer.)
Woodin's result is that if $\mathrm{lh}(u^j)=\kappa^+$ and $2^\kappa=\kappa^{++}$, then $\mathbb R_u$ preserves the inaccessibility of $\kappa$ and forces the failure of $\diamondsuit_\kappa$. Stronger large cardinal properties are preserved, under stronger assumptions on the length of $u$, and the same argument gives failure of $\diamondsuit_\kappa$ in the extension: If $\mathrm{lh}(u)=(\kappa^+)^2$, then $\kappa$ is Mahlo after the forcing, and if $\mathrm{lh}(u)=(\kappa^+)^{\kappa^+}$, then $\kappa$ is still greatly-Mahlo.
Unfortunately, it is not known how to preserve weak compactness this way (the known bounds on the length are much worse, we do not know if $\mathrm{lh}(u)=2^\kappa$ is an upper bound, for example; what is known is that preservation happens before we reach the first repeat point). Anyway, the argument for the failure of diamond breaks down if the length is too long.
That being said, let me add that Hugh's result gives more than the failure of diamond: In fact, he shows that no candidate sequence in the extension can guess every subset of $\kappa$ from the ground model. (And, modulo the technicalities of Radin forcing, the argument is surprisingly short.)
You should contact James, as he will know for sure whether there are further developments on the problem for weak compactness, and whether there are improvements on the upper bounds on the known cases.
Best Answer
I suppose there is some interest in propositions implying large powerset. For example, a classical question at the infancy of set theory and real analysis was whether there is a probability measure on the unit interval that extends Lebesgue measure, in which every set is measurable. This implies that the continuum is weakly Mahlo, Rowbottom, etc. It's also equiconsistent with a two-valued measurable cardinal.
The stronger interest in large cardinals is surely due to the empirical fact that they form a coherent system that measures the relative consistency of everything. It's not clear that large powerset type propositions have a similar property. Some of the things you mention have no strength, like the continuum function is injective. It follows from GCH and it's negation is easily forced.
Related fact: If $2^\omega$ is real-valued measurable, the continuum function is not injective. $2^\kappa = 2^\omega$ for all $\kappa < 2^\omega$. This is due to Prikry, and a proof can be found in chapter 22 of Jech.