I have never really read anything proper about model theory, so I have a few questions:

Someone told me that a school of logicians managed to give a very short proof of Falting's Theorem using model theory (and apparently "elimination of quantifiers"); I have not been able to find any reference about this, any ideas?

I also remember being given some very interesting information about the model theory of some rings. I do not recall the specifics, but it did give precise insights about how something like the complex numbers is easy to work with, whereas for the integers it is much harder (with problems such as Matiyasevich's Theorem/Hilbert's Tenth Problem). I have not managed to find anything similar to what I remember about it.

Does anyone have any information about all this, or a good source to learn from (which would, hopefully, mention these examples, at least the second one as it seems like it must be fairly standard)?