Train a GAN on "before and after" images of dental surgeries
It's a very specific problem and there's no right or wrong solution.  I'll just write what I'd do in your position and hope that it is useful.


  How many "before and after" images will I need?


You'll need a lot of images to consistently get good results, in the range of tens or preferably hundreds of thousands. 


  What architecture should I use? i.e. StyleGAN, VAE?


StyleGAN seems to be the most popular option for these sorts of tasks these days, so I'd stick with this.


  As I will need my custom model; training a fresh model would need a good GPU. What platform should I go for? Colab, Floyhub, or something else


The platform you'll choose depends on whether or not you're willing to pay. Colab, Floydhub and Kaggle are the only free options with GPU afaik, but also offer paid versions (1, 2). All have their pros and cons. For example colab, which I'm more familiar with, offers only 12-hour sessions (after that it disconnects you and you have to reconnect).


  Approach 


Regarding the process you said, that's an interesting approach. I would have gone with something more basic like a segmentation mask, but I think your idea is better.