I've got chronic pain from back problems, which includes nerve problems, myopathy, lymphedema, and symptoms of hypermobile EDS, just to name a few of the medical conditions I've got, and I've got a few mental health issues. (It's a ridiculously long list, but I can go into more detail if needed.) Plus, I had blood clots several years ago with no determined reason, and I'm on T and plan on getting surgery after I get better insurance. The injections are trickier for me because of potentially aggravating one of my medical conditions.
Only a doctor who's familiar with your history can really tell you what you need to watch for and if it's safe for you to start T. I had to get a letter from a therapist before I could start T, and I'm probably being monitored a little more than most people, but so far things have been going well.
I should mention that I decided that for my mental health I had to transition and made an informed decision to take the chance that transitioning may aggravate one of my medical conditions. My dysphoria had reached a point where I couldn't continue not doing anything. I also got lucky with having to find a new PCP, and the one I saw had friends who worked at a trans friendly clinic, so I basically have two groups of doctors monitoring any changes that happen.