In what sense are loop diagrams quantum corrections?

The reasons were given here. Essentially, at tree level you recover classical results. Loop corrections are proportional to powers of $\hbar$ and these are quantum terms.

Tree level diagrams are one-particle relativistic quantum mechanics, but not quantum field theories.

The point is obscured by two reasonable details in modern QFT books: they avoid to speak of "2nd quantisation", and they set h=1 everywhere (so for instance it is not so clear how different the h->0 limit is for fermions than for scalar fields, or how bosons can add to build a classical electromagnetic field but fermions can not). It is expected that you go across a relativistic quantum mechanics textbook before jumping to QFT, but sometimes the career path is different.