If gender identity were 100% arbitrary social construct, would there be any transgender people?
Absolutely. Being trans has little to do with how other people perceive me, and almost everything to do with how I perceive me. This is why Gender Dysphoria is the description in the DSM, and why the treatment is to have Hormone replacement therapy, and then surgery.
When I look in the mirror, i see things that my brain says are wrong. my facial hair is wrong, my penis is wrong. They are not supposed to be there. It feels chemicals in my body that it says are wrong.
Having Hormone replacement therapy helps, not just because of physical changes, but also because my hormone levels are closer to what a cisgender woman would be. If anything, the physical changes are sort of secondary? they help with the mirror, and they help with people calling me what I'd like to be called, which are both good for the brain. When I've had to be off hormones for extended periods my mental health has been much worse generally, even though the physical changes have not gone away.
I guess a lot of that also goes to help answer your question 1. Basically, when I'm on HRT to change my body and to alter the levels of estrogen and testosterone in my body, I hate myself a whole lot less. I hate my body a lot less, and it's easier to function - It's like having a block lifted out of the way. I can focus on proper diet and exercise, I don't just get up, go to work, go home, eat, sleep, which is what my existence was by the time i finally got into the therapy step before HRT,, then some time later on to HRT.
As for 2.), well, your problem is that you're looking for biological stuff towards Gender, and most gender stuff is brain related, and unrelated to the rest of your biology. In fact, one of the biggest "theories" for why transgender people exist has to do with a testosterone surge in the intrauterine period of development. Basically, XYs get a shot of testosterone which affects their brain in some ways, and XXs don't. except, there may be times when XYs don't get that "shot" of T, or XXs do, or the amount of testosterone in the "shot" is not as much as it should be. All of these cases have the potential to create gender variance inside of XX and XY people.
If you take that hypothesis and throw it at the scientific data, you come up with this: Male-to-Female Transsexuals Have Female Neuron Numbers in a Limbic Nucleus. If you read the abstract, you can see that the opposite is also true, that Female to Male trans people have Male Neuron numbers.
Does that count as a biological basis for gender Identity? I'd like to think so, I guess, but again you have to focus on brain and endocrine science in the literature to get anywhere.