Dear Claude, you are extrapolating electromagnetism way too high. You're going from low energies to the Planck scale, assuming that nothing qualitatively changes, but this assumption is wrong.
The fine-structure constant is essentially constant below the mass of the electron - the lightest charged particle - which is 511,000 eV or so. You are extrapolating the running of the electromagnetic fine-structure constant $\alpha = 1/137.03599$ all the way up to the Planck scale, about 10,000,000,000,000,000,000,000,000,000 eV. I chose to avoid the scientific notation to make it more explicit how far you have extrapolated.
However, at the electroweak scale, about 247,000,000,000 eV, which is much lower than the Planck scale, the electromagnetic force is no longer the right description. The weak nuclear force gets as strong and important as electromagnetism and in fact, they start to mix in nontrivial ways. The right theory doesn't use $U(1)_{\rm electromagnetism}$ but $SU(2)_{\rm weak}\times U(1)_{\rm hypercharge}$. Note that the electromagnetic $U(1)$ is not the same thing as the hypercharge $U(1)$.
So instead of the fine-structure for the electromagnetic $U(1)$, one must express physics in terms of the fine-structure constants for the new $SU(2)$ and $U(1)$ electroweak gauge groups. These fine-structure constants are not as tiny as the electromagnetic one.
The hypercharge $U(1)$ fine-structure constant gets stronger as the energy grows - much like the electromagnetic one would - while the $SU(2)$ fine-structure constant gets weaker as the energy grows (however, it would be also getting stronger, like electromagnetism, if we included new $SU(2)$-charged particles such as superpartners).
At the same time, the $SU(3)$ QCD fine-structure constant is getting weaker as we raise the energy. Ultimately, all the three fine-structure constants, when properly normalized, become close near $10^{16}$ GeV which is the GUT scale - they exactly cross if one includes the superpartners of the known particles.
The common value of all these three fine-structure constants at the GUT scale is something like $1/24$ or $1/25$ - in fact, there exist somewhat preliminary arguments based on F-theory (in string theory) and similar frameworks that suggest that the number could be exactly $1/24$ or $1/25$ when certain conventions are carefully followed.
If your question was not one about the real world, but one about a fictitious world that only contains QED up to the Planck scale, then indeed, the fine-structure constant would increase just "somewhat", perhaps to $1/100$ or so; indeed, if the world were pure QED, the Landau pole with $\alpha=\infty$ would occur at much higher energies than the Planck scale.
The number can be calculated as a function of $\alpha_{E=0}$. However, it is meaningless to assign error margins to a theory with adjustable parameters that doesn't describe the real world - and QED above the electroweak scale doesn't. A theory that doesn't describe the real world is always in error, even when you're within any error margins, and you can't just "match" it to the real world because different ways of "matching" two different theories would yield different results.
Answer to the main question:
It is a well regarded fact that the terminology unified electroweak interaction is a bit of an abuse of terminology. What the term means is that both Quantum Field Theories, the Hypercharge ($U(1)_Y$) and Weak ($SU(2)_L$), are unified in a common framework, which predicts the low energy electromagnetism ($U(1)_{em}$) through the Higgs mechanism
$$ U(1)_Y \times SU(2)_L \to U(1)_{em}$$
It does not refer to an unification scenario as in the Grand Unified Theory (GUT) setup, it's meant to refer to an unification in the sense Weak decays and Electromagnetism are understood as remnants of a higher-energy theory, the Standard Model.
In comparison with GUTs the terminology can be applied if you think that regular GUT setups predict unification into a gauge group which is composed of solely one (semi) simple Lie group, e.g. $SU(5)$, $SO(10)$ being 2 of the most popular. In this sense the couplings do unify. The electroweak unification can be regarded as a unification into a group with 2 (semi) simple Lie group factors, the $SU(2)$ and $U(1)$. It is in this way of thinking about it that people refer to as unification. Notice that in the later case each factor can has its own coupling, and so the couplings are not equal, i.e. do not unify.
Answer to the bonus question:
What you asked is a big open question. Fermionic masses come from Lagrangian terms called Yukawa couplings, for example for the electron
$$ y H L e+\mbox{h.c.}$$
for example, and the masses are then
$$ m \sim v y$$
where $v$ is the vacuum expectation value of the Higgs field, $H$, and $y$ are the Yukawa couplings are not specified in the Standard Model and one should expect them to be of order 1. But this only happens to the top quark, while all the other fermions have a lot smaller (in some cases many orders of magnitude smaller) than 1. Why this is like this is still an open question in Physics.
Best Answer
There is of course no unique answer because if the normal GUT embedding and theory is wrong, there may still be another valid theory that naturally involves the same or (almost) any other rescaling of the hypercharge fine-structure constant.
If you say that GUT is wrong, it is far from specifying what is actually right. When it comes to coupling constants and other things, GUT is more predictive than theories without any unification - that's a characteristic feature of unifying theories. So if you say that GUT is wrong, you just reduce the ability to predict and explain patterns in Nature, so you can't expect that by this negative assumption, you will obtain positive predictions. One shouldn't expect that by throwing out theories that are meant to explain something, one automatically finds different answers to the questions.
Of course, if you assume a normalization with no relationship between any groups where the hypercharge $U(1)$ is normalized just like any other $U(1)$ would be, and your goal is to make the formulae as simple as possible on the paper (which is not really a physical criterion), then $5/3$ is replaced by $1$. You just omit the $5/3$ factor. But this is a kind of vacuous statement because one may only compare the fine-structure constants of the different group factors if there is some relationship between them which is either grand unification or plays the same role.
One more preemptive comment: at low energies, it is not true that the hypercharge fine-structure constant renormalized by another simple factor such as $5/3$ yields the electromagnetic fine-structure constant. At the GUT scale, similar relations exist but the electromagnetic fine-structure constant is not well-defined there.