Type 1, High TSH, low testosterone

You didn’t say what your total testosterone (T) was, but generally T is free (FT) when it is not bound by sex hormone binding globulin (SHBG). SHBG can increase when you are losing weight, as this increases SHBG, so reduces FT. As for its effect, it’s questionable. My guess, is that unlike Type 2, which in men can correlate with lower T, FT, and SHBG, Type 1’s are continually calorie restricting, and might have other anomalies like elevated albumin/globulin, although these ideas are purely speculative, but normal T with reduced FT.

I have heard doctors say it is just a calculation, and so dismiss it. I have read many studies looking at mens sexual function, sometimes in relation to diabetes, but generally, the study results are not consistent. Sometimes T has a positive correlation with sexual function, while FT does not, and sometimes it is the reverse. I think that a meta-analysis might show both to have a positive correlation with sexual activity in men, but I have not seen a study like that, and even then, it might be a limited relationship and not strong.

I’ve seen few studies on women in that regard, so I can’t write much, but the effects might be more pronounced, not less. Men’s T is about 40 times as much as women’s, but much of it is bound, while in women the effect of T and FT might be more pronounced. Again, I’ve seen an inconsistency in studies.

Yes the tsh is generally used as an indicator.
You doctor should run a full thyroid panel including
T4
Free T4
Tsh
Tuptake ( not done too much anymore) but it think it’s helpful to get the full picture.
T3 total
Free T3
If all the others are normal, I wouldn’t worry about the tsh.
The thyroid system is complicated with constant checks and balances.
Tsh stimulates the thyroid to make more T4.
So when it is high, it could mean your thyroid isn’t making enough T4.
But you should have it checked.

This will makes wt loss hard for sure. As for diet I’d say lose the Soy, gluten and dairy for sure. I would also make sure you don’t use fluoride it negatively effects thyroid. Perhaps speak with a naturopath. I started taking a supplement and my levels popped back up. It’s called GTA if you wanted to research it.
I was extremely depressed, couldn’t control my weight and had such low energy. Hopefully that’s not the case for you. I’m still working through it.