For the name thing, make 100% sure that every account you have knows that you changed your name. Banks, student loans, car loans if you have one, etc. in addition to all the usual suspects like the DMV and Social Security. Also if you are employed make sure you're putting in new paperwork with your job.
Essentially the old name will linger on your credit report, taxes, etc. until it no longer exists anywhere. IME, it takes about a year from the time you changed it on the last account for it to completely disappear.
If you are having trouble with the birth certificate, consider just getting a passport. With an updated driver's license and a letter from your doctor, the State Department will allow you to update it. It is used for the same purposes as a birth certificate. It's what I encourage people to do in states where you can't amend a birth certificate.
For health insurance, this is tricky. The advice I was given was to leave insurance as female until I no longer had any parts that would require female preventative care. So no mammograms, no pelvic exams, etc. BUT I decided not to go that route, and updated it to male after my top surgery. I had no major issues getting my hysterectomy covered. They had to make some minor coding adjustments in the system to have it accept the claim, but it eventually got straightened out.
But this was in Washington DC where insurers legally can't not provide transgender healthcare coverage. My plan is also an ACA plan, and a provision of the ACA removes gender coding from procedures. I am not a legal expert, so I can't tell you if this only applies to plans offered by the healthcare marketplaces, or if it applies to all plans. I feel like Minnesota is a very trans friendly state in terms of medical stuff, so I would encourage you to reach out to local resources there and see what the consensus is.
