I will say that I actually /gained/ weight when I started T. Some was muscle, some was fat. But I gained because I was eating a lot of takeout and drinking soda and other things with hella sugar and fat. So, my metabolism /may/ have increased some, but I was eating too much crap, which kinda negated any metabolic boost T may have given me.
Not to say you should NEVER eat that sort of stuff. I was just eating it all day, everyday, and since I gave up the takeout and the soda, I dropped a good 25 pounds, so I'm back to my pre-T weight, although I still have more muscle than I did before that, so I feel okay about it. I still drink fruit juice on the regular. I also have diet soda and ramen a few days a week (at least), so my diet isn't perfect. And I have sugar and fat in my diet, of course; just not as much as before.
As far as your metabolic rate goes, there are some blood tests your doctor can and should do periodically anyway while you're on T. I'm horrible and haven't had any blood work done since I started T, because I don't have the money. I barely have the money to get my T vials, but that is part of why I cut out the more expensive and unnecessary foods I mentioned. Empty calories and v. expensive. But I digress, again.
I don't have the spoons to research it at the moment to back myself up, but you may ask your doctor to do a thyroid panel, if they don't suggest it first. Thyroxin and other thyroid hormone levels can give some info about how your metabolism is doing. Your doctor may also advise you to have other blood work done to check on it.
As for body temperature, I definitely notice I feel hotter. I haven't taken temperature readings, so I obviously can't say for sure that my internal temp has significantly increased, but I will say that it certainly does feel that way. Summer is barely even beginning, and I'm dreading it. (I live in Florida, so like... it's gonna be interesting.)
EDIT: As far as exercise, nothing significant. I take walks when I have the spoons for it, but I've got bipolar disorder and it can be tough for me to get out and do stuff like that. I actually really miss running, so I hope I'll feel better in the future and be able to do that again, because I'm super interested in seeing whether my performance there will change, if I get back into it steadily.