Wouldn't this run afoul of the ODR, anyway?

I don't think so, ODR applies per program, the library and main program are separate.

Object layouts will be likely different and the main program will probably crash.

Not so: The Standard doesn't have anything to say about the differences between static and dynamic libraries, or the differences between a .exe and its .dll's (or .so's). The program is composed of all the things that link into it as far as the ODR is concerned.

The breakage from mixing different library versions (say, MSVCRT versions, for example) is a direct result of violating the ODR.

The standard doesn't say anything at all about dynamic linking AFAIK, so it's basically OS-dependent. I think it works (as in the algorithm is well defined) in Linux, but it's best avoided due to complexity and non-portability.

Exceptions or STL classes shouldn't cross library boundaries though, otherwise things will break.

It will work perfectly fine as long as the libraries use compatible ABIs.

Make sure that your compiler provides a stable std library ABI.

Good point that it's technically doable.

Is this done a lot in practice? I had the impression that a lot of projects stick to C APIs in dynamic libraries.

MSVC does not guarantee a stable standard library ABI, so those that need to target it are out of luck. In linux land, the switch from libstdc++5 to 6 still haunt the memories of many, although it was quite a long time ago (gcc 3.4) and the ABI has been stable since.

In practice the reason that many projects provide a C ABI is that they want to interoperate with other languages and the C ABI, being extremely minimalist, is the least common denominator.

And that's why exceptions have no real world usage in C++, except for a few local functions here and there.