Does Gödel's theorem show that we can't prove anything in mathematics?

Gödel's theorem is sometimes invoked in support of skepticism, either with respect to mathematics or about reason as a whole. The following quotations illustrate:
Goedel's Incompleteness Theorem guarantees the inability to have absolute knowledge.
Indeed, a long list of philosophers, among them Sextus Empiricus and Kurt Gödel, have PROVEN that the foundation of reason cannot be confirmed. It may be difficult to believe, but logic is proved invalid by its own standards.
Most statements in math cannot be shown to be true... This is a consequence of Godel's theorem.
It's usually unclear just how the skeptical conclusion is supposed to follow from Gödel's theorem. To make the matter more concrete, let's consider a theorem of mathematics, Dirichlet's theorem:
If k and l are relatively prime positive integers, then the arithmetic progression l, l+k, l+2k, l+3k, ... contains infinitely many primes.
That k and l are relatively prime means that they have no common divisor except 1. Thus Dirichlet's theorem generalizes Euclid's theorem that there are infinitely many primes. Not only are there infinitely many primes in the positive number series 1,2,3,4,.., but there are also infinitely many primes e.g. in the series 3,8,13,18,..

Now let's consider whether it is somehow a consequence of Gödel's theorem that we don't have absolute knowledge of the truth of the theorem quoted, or cannot show it to be true, or that any proof of the theorem must be somehow flawed.

Note that although there may be any number of statements that we can't prove, of which we can't have absolute knowledge, and so on, the issue is whether we have proved, have absolute knowledge of, etc, this particular statement. Thus it's not enough to claim that Gödel's theorem shows that there are true statements that cannot be proved, if we wish to draw a general skeptical conclusion from the incompleteness theorem.

It would appear that any support for a general skeptical conclusion to be gleaned from Gödel's theorem must refer to the second incompleteness theorem rather than the first. That is, the formal unprovability in T of the consistency of T is what is thought to undermine theorems like the one quoted.

Suppose we have formalized the part of mathematics used in proving Dirichlet's theorem in a formal theory T. The consistency of T is not provable in T, although Dirichlet's theorem is. How does this affect whether or not we know that Dirichlet's theorem is true, or whether or not the proof the theorem is conclusive?

It appears that many people who regard Gödel's theorem as supporting skepticism reason as follows:

If we can't prove that T is consistent, then no theorem proved in the theory T can be regarded as proved (known to be true). For the theory may be inconsistent, and then a proof in it proves nothing.
For a perspective on this, first note that we cannot expect to prove everything in mathematics. There must be axioms or basic principles or methods of proof which we do not prove correct in any mathematical sense, but accept without mathematical proof. So suppose we accept the axioms and methods of proof formalized in T as valid without proof. How is Gödel's theorem relevant in criticizing such a view of the axioms and methods of proof of T?

Note that I'm not arguing here that we do have absolute knowledge in mathematics, or that we are justified in accepting the methods and axioms of T without proof. The question is only whether Gödel's theorem implies that there is anything wrong with this.

Now Gödel's theorem implies that the consistency of T cannot be proved in T. Why should this be an argument against our accepting the axioms and methods of T? After all, we already know that not everything in mathematics can be mathematically proved. We must accept something without mathematical proof. Gödel's theorem doesn't tell us anything we didn't know in this regard.

Also, what if the consistency of T were in fact provable in T, contrary to Gödel's theorem? Why should this have any effect on any skeptical doubts we may have about T? After all, a consistency proof in T would be just another proof in T, itself open to doubt.

Finally, consistency of T is quite insufficient to guarantee that our proof of Dirichlet's theorem is correct. Logically, it is perfectly compatible with "T is consistent" that (i) "there are infinitely many primes in the series 3,8,13,18..." is provable in T, and (ii) there are only finitely many primes in the series 3,8,13,18.... So any doubts one may have about the validity of the methods and principles formalized in T as used in the proof of Dirichlet's theorem would not be swept away by a consistency proof for T.

In summary: Gödel's theorem does not imply that there is no justification for accepting the axioms and principles of T without mathematical proof. Nor does mere consistency of T imply that those axioms and principles are valid. There may be good arguments for general skepticism with regard to mathematical proofs, or logic, or reasoning, but Gödel's theorem does not provide such.