It really depends on individual circumstances. In general, what I have experienced and heard other folks experience is as long as you are actively managing your conditions or they are related to your dysphoria, you will have no issue getting HRT. If it's something that you're not addressing and it isn't related to dysphoria, you will probably need to make steps towards managing it before a doctor will agree to prescribe. Doctors in general feel better referring you for HRT or surgeries when they know that "your house is in order" so to speak - any pre-existing conditions are known and managed, you have a solid support system, you are well informed, etc.
I had a whole slew of mental health issues pre-everything (social anxiety, generalized anxiety, depression, BPD). When I visited a doctor to start testosterone via informed consent, we discussed my issues and both mutually felt that dysphoria was likely the root cause of the majority of my feelings. My anxiety was all related to how people perceived me, and being put in situations where I had to be "female". I was depressed because I had known for years that I wanted to transition but kept putting it off in hopes that I would outgrow it or family would become more accepting. All the symptoms of BPD I had were related to self image, mood, disassociation, and self mutilation - easily explainable when you consider dysphoria.
I spoke with 2 different therapists while getting referrals for top surgery, which was around the same time as the above, and they both agreed that all of the issues I had experienced thus far were very likely stemming from dysphoria. I started T around the same time that I was speaking with both, and felt better fairly quickly. My anxiety started to go away as I started being gendered correctly in public, and was completely gone after my voice had dropped a few times. The only time it flairs up now is before doctor visits related to bottom surgery, because that involves me getting undressed from the waist down. I'm not depressed anymore. I have a stable self image now, my mood is consistent, I no longer feel the need to disassociate or self mutilate. I saw a new therapist and psychiatrist recently for bottom surgery referrals and they were both very pleased with what my transition had done for my mental health, and were happy to refer me for bottom surgery.
I had taken medication for a lot of these in high school and college, but stopped a few years ago. I did do some DBT sessions for BPD, which I did find helpful. But T has largely taken away any need to go back on medications for mental health reasons, so I'm not sure if there are issues with taking both. I imagine it would depend on the specific medication. You do need to stop some medications prior to surgeries at the surgeon's request, but again, it completely depends on what the medication is and what the logic for stopping would be. I had to stop T prior to top surgery due to bleeding/clotting risks, but not before my hysterectomy because they didn't want me to be without hormones.
