Well, I am wrong person to advice anything about medicine :)

From practical point of view, I would recommend go through fast.ai courses
http://course.fast.ai/part2.html where they kind of answer to your questions 3 and 4.

Regarding 1 — use SGD-like. Take some problem, which you can find, e.g. from kaggle or NIPS workshop on medical deep learning research and start working + reading paper on it. But don`t stick to first one too much — change directions to more important or interesting things when you find them. Repeat until convergence :)