I've looked into this a bit myself, because I'm getting towards that stage in life. I had been under the impression that testosterone levels drop sharply after menopause, but it turns out that's not the case. T levels do get lower throughout a woman's adult life, so a woman in her fifties and sixties will have much less than a woman in her twenties, but that's regardless of her menopausal status. At least that what I understand from my limited research. Someone can correct me if I am mistaken.
As far as being hot or cold, I'm not sure how that would affect training. I've had a few hot flashes before, and I suppose if it happened while I was training, I might want to sit and wait for a moment until it subsided, just so I'm not distracted by being all prickly and hot while I'm trying to concentrate on my squat. Personally I find it's best not to overthink these things.
That's just me, though, and I'm no expert. I do hope that there will be MUCH more research and serious articles about older women who strength train, though. Menopause or no menopause, a lot of us want to get strong without having to wade through condescending twaddle about doing presses with soup cans.