How to avoid personal issues interfering with my Ph.D. studies?
The problem is easier than it seems. Your first responsibility is to yourself. What you most need to do is find a way to build a life for yourself that you will enjoy over the long term. This could involve a PhD or not, however, you seem to really be attracted to Academia where a doctorate is required. Earning the degree might actually be the sacrifice you make to achieve your goal. But keep the "good life" rather than the PhD foremost as your goal. 

A number of things in what you say indicate that you might profitably look at a different subfield than mathematical biology. You can use your current position, as well as your past education, as a base from which to conduct your search, speaking with other potential supervisors and even institutions. Don't underestimate the value of a good supervisor, of course, and she may be able to help you if you approach her in the right way. 

Having been chosen, rather than choosing is possibly a big part of your problem. You "went along" at a point at which it might have been better to "take charge". This led to a number of your other dissatisfactions, I think. But you can take charge without making a clean break at this point until you see something that you really want to do. 

Also, since you were chosen you should have confidence that your skills are not in doubt.