How true are theorems proved by Coq?
For systems like Coq that are based on type theory, this question is trickier to answer than you might expect.
First of all, what does it take to "know" the consistency strength of some system? Classically, the most thoroughly studied logical systems are based on first-order logic, using either the language of elementary arithmetic or the language of set theory. So if you are able to say, "System X is equiconsistent with ZF" (or with PA, or PRA, or ZFC + infinitely many inaccessibles, etc.), then most people will feel that they "know" the consistency strength of X, because you have calibrated it against a familiar hierarchy of systems.
Coq, however, is based on something called the Calculus of Inductive Constructions (CIC). Without going into a detailed explanation of what this is, let me just mention that the core of CIC doesn't have any axioms, but typically people add axioms as needed. For example, if you want classical logic, then you can add the law of the excluded middle as an axiom. To get more power you can add more axioms (though you have to be careful because certain combinations of axioms are known to be inconsistent). But trying to line up the various systems you can get this way against more familiar set-theoretic or arithmetic systems is a tricky business. Typically, we cannot expect an exact calibration, but we can interpret various fragments of set theory in type theory and vice versa, showing that the consistency of CIC plus certain axioms is sandwiched between two different systems on the set-theoretic side. If you want to delve into the details, I'd recommend the paper Sets in Coq, Coq in Sets by Bruno Barras as a starting point.
The folklore result is that there is a "simple" model of the underlying theory of Coq in $\mathrm{ZFC}+\omega$-many inaccessibles.
A good intro to this model is "The not-so-simple proof-irrelevant model of CC" by Miquel and Werner.
Benjamin Werner also sketched a consistency proof for the more general system with universes: On the strength of proof-irrelevant type theories. I feel that the community is begging for a clean formal construction of this model. I believe that this is what Bruno Barras is doing (as mentioned by Tim Chow in his answer).
The lower bound is more elusive still. I'm only aware of 2 results for lower bounds:
Werner again builds an interpretation of ZFC in CIC + some set-theoretically plausible axioms. Sets in Types, Types in Sets.
Miquel (again) builds an interpretation of Zermelo Set Theory in $\mathrm{F}_{\omega^2}$, the non-dependent fragment of $\mathrm{CoC}$ with universes. The article is a must read: $\lambda Z$: Zermelo’s Set Theory as a PTS with 4 Sorts.
If memory serves, in his PhD, he shows the more general result that $\mathrm{CoC}$ with universes is equi-consistent with Zermelo theory with $\omega$-many (Zermelo) universes. Sadly the dissertation is in French.
I'm far from certain that $\mathrm{CIC}$ with universes is more powerful than $\mathrm{CoC}$ with universes if no additional axioms are added. This intuitively is because inductive data-types can be built at the "2nd level" using the usual encoding trick, and AFAIK, have the same strength as "built-in" inductive data-types (but are significantly less convenient to use).
As you can see, there is significant work to be done to put these questions to rest.
Here are some publications related to your question:
Robert Pollack. How to believe a machine-checked proof. In G. Sambin and J. Smith, editors, Twenty Five Years of Constructive Type Theory. Oxford Univ. Press, 1998.
Pollack-inconsistency, Freek Wiedijk Freek demonstrates the most popular proof assistants are Pollack inconsistent.
In an internet post Pollack discusses Coq coercions:
The problem is that Coq coercions are informally specified and behave somewhat unpredictably. A formal theory of coercions, such as Luo's Coercive subtyping (with proof theory and semantics) would eliminate this question of the meaning of statements using coercions. However, the proof theory of coercions is complicated.