Regarding this part of the OP, as a CIS gendered person, I would like to say I hear ya’. It is very confusing and unintuitive to us because since our gender and sex are in alignment we can’t easily separate the two. As I said in the OP of another thread (which may be of interest to you regarding non-binary), trying to explain transgenderism to a CIS gendered person seems a bit like trying to explain drowning to a fish. They can’t understand because they have never experienced not being able to breath water.
Gender is some ineffable thing that tends to be correlated with but is not identical to all of the various things you listed. But whatever it is it is very important to peoples psyche such that suppressing it can lead to depression and suicide, and telling people to get over it and accept that their birth gender is who they are is about as effective as telling the drowning person to just stop struggling and breath the water like any normal fish.
Fortunately you don’t have to fully understand transgenderism, all you have to do is accept it. Accept that they exist, that what they are expressing is real, and that there are certain things that we take for granted, that don’t apply to them. Then put in the minimal effort to try to act in a way that will acknowledge this understanding. If you screw up, apologize. Chances are that will be the end of it. If they bite your head off, it could be either that they are mistaking you for someone who is going out of their way to make them uncomfortable, or it may just have been a long day/week/life and you are the straw that broke the camel’s back. In any case a sincere apology with the effort to do better in the future is the best way to defuse the situation.