If I have studied French in school, it means I spent time trying to gain knowledge about French. It doesn't mean I can speak French now.
Correct. Many people study French for a year or two or even longer in school and yet don't feel confident speaking it in adulthood. In some cases it may have been a mandatory course - one in which they had no real interest.
If I can speak French now, if I know French, I have learnt French.
Correct (although depending on context, "I have learnt French" might be interpreted as meaning that I have only recently finished learning it - so "I know French" or "I speak French" might be better).
If I have studied French at university, does it mean I have a degree in French?
Not necessarily. It is certainly one possible interpretation, and perhaps the most obvious one. But you may have studied just a few modules in French as part of a degree that is mainly in a different subject.