While I will answer your raised concerns one-by-one and it's perfectly possible to get your project reviewed, I'm not convinced Code Review is your best bet.
Does Code Review allow for posting (or linking to) entire projects?
Posting? Yes. Provided it fits within our character-limit of a post. The code must be in the post itself. We don't review entire repositories behind a link.
There is a hard limit of 65,536 characters per post. Within that will have to fit the code, a description of what the code is supposed to do and all the other things relevant for us to provide a review. See Simon's guide on writing a good question.
I'm not sure if this helps at all, but in my case, I am not looking for generic input, but rather I have a specific question about how to improve an aspect of my program.
If you have a specific problem, focus on the specific problem. However, if you already know the problem, do you still need a review pointing out your problem? If you have a specific problem requiring a specific solution, you're a step beyond review already. A review might give you a lot of good advice while not fixing your immediate problem.
Perhaps Stack Overflow or Software Engineering are better places to look for solutions, depending on the problem you're facing. If the problem is that the code doesn't produce the correct output, it's not ready for review.
Coming from Stack Overflow, I know about the concept of a minimal reproducible example, but I am not sure if that applies here.
No, it does not apply here. Minimal Complete Reproducible Examples (MCVE) are something required on Stack Overflow while outlawed on Code Review. To quote our help center: "Details matter! In order to give good advice, we need to see real, concrete code, and understand the context in which the code is used."
If anything remains unclear, please leave a comment.