As has been mentioned, there's no homotopically meaningful content to the notion of an isofibration, since every map of $\infty$-categories is equivalent to an isofibration. So the point is really all in the definition of an $\infty$-cosmos: isofibrations are the kinds of maps between $\infty$-categories that you can take a strict pullbacks along, or take the strict limit of a countable tower of, and so on.
It is right at the heart of homotopical category theory that we often want a strict construction to model a more complicated, homotopy coherent construction, such as a homotopy pullback, basically because this saves us from carrying around lots of coherence isomorphisms in our arguments, replacing them with equalities. This can't be done for pullbacks along arbitrary $\infty$-functors, but for isofibrations, it works, which allows an $\infty$-cosmos to behave something like the category of fibrant objects in a simplicial model category. If you don't have any familiarity with model categorical arguments, then you'll get some as you read further in the book and see how the strict limit properties of isofibrations enable many of the arguments Riehl and Verity make. But in particular, don't worry too much about isofibrations, which are just a technical convenience–keep reading to get to the good stuff!
For any symmetric monoidal $\infty$-category $\mathcal C$ and commutative algebra $A\in \mathcal C$ (you're supposed to think of $\mathcal C$ being $Cat_\infty$ itself, or maybe $Pr^L$; and $A= QCoh(Spec(k)) = Mod_k$), there is an equivalence: $CAlg(\mathcal C)_{A/}\simeq CAlg(Mod_A(\mathcal C))$, where the right hand side is to be interpreted as you expect if $\mathcal C$ has enough appropriate colimits, and can be made sense of operadically in general).
This in particular means that if you have a diagram $A\to B\to C$ of commutative algebras, then $B,C$ can be made into $A$-modules canonically (in fact into commutative $A$-algebras, but you can forget most of that structure), and the map $B\to C$ can be made $A$-linear canonically.
You can apply this to $QCoh(-)$ applied to $X\to Y\to Spec(k)$ and get a canonical $Mod_k$-linear structure on $f^*:QCoh(Y)\to QCoh(X)$. By naturality, this coincides with the approach you describe in (1).
Now, given a symmetric monoidal $\infty$-category $A$, and an $A$-linear functor $f: M\to N$, if $f$ admits a right adjoint $f^R$, then $f^R$ is canonically lax-$A$-linear - this follows from the theory of relative adjunctions in Higher Algebra, but concretely boils down to the existence of (coherent, natural, compatible etc.) maps $a\otimes f^R(n)\to f^R(a\otimes m)$, which are simply the usual projection maps, the mate of $f(a\otimes f^R(m))\cong a\otimes ff^R(m)\to a\otimes m$, where the first equivalence is the $A$-linearity of $f$, and the second map is the co-unit of the $f\dashv f^R$-adjunction. It then becomes a property for $f^R$ to be $A$-linear, namely the property that these projection maps be isomorphisms.
A cool thing to note is that this projection map is always an isomorphism when $a\in A$ is dualizable. So in your situation, $f_*$ is always $Perf(k)$-linear. If furthermore $f_*$ preserves colimits, then because $Mod_k$ is generated under colimits by $Perf(k)$, it follows that $f_*$ is $Mod_k$-linear; and this is essentially an "if and only if" (if $f_*$ is $Mod_k$-linear, one can prove that it preserves arbitrary coproducts and thus arbitrary colimits). This happens quite often, e.g. if $X,Y$ are qcqs, but presumably more generally too.
I'm not sure what $f^!$ is in an algebro-geometric context, but pesumably some similar game can be played there.
Best Answer
Oops, this is actually not hard, just using 1-categories. Explicitly, take two maps from the terminal category to Ab, one landing in $\mathbb{Z}$, one landing in 0, both are lax symmetric monoidal. The pullback (after replacing the first map by an equivalent isofibration but we can mostly ignore this technicality) is empty so not a symmetric monoidal $\infty$-category (where it would be an ordinary category anyways). So indeed, symmetric monoidal $\infty$-categories with lax symmetric monoidal functors do not form an $\infty$-cosmos.
Though, if we are pulling back $(G,F):C\to A\times B$ along a two-sided fibration $(p_1,p_0):D\to A\times B$ from $A$ to $B$ (which is to say, $p_1$ is cocartesian, $p_0$ is cartesian and lifts are compatible), where $p_1,p_0,$ and $F$ are symmetric monoidal, while $G$ is lax symmetric monoidal, this should work. This case can be used to show that cyclotomic spectra are symmetric monoidal, for example.