No. You can be completely exerted without sweating or completely soaked in sweat without feeling exerted at all. I’ve experienced both and when I’m in better shape I will sweat more without feeling exerted, while when out of shape I can feel exerted without being sweaty. The amount of energy required to move your body is about the same whether you are fit or not. So the heat production and therefore the sweat production would be the same. Being in better shape makes the body better at producing sweat to cool you down. So more sweat and less exertion.

How fit you are doesn’t determine how much you sweat. Becoming more fit can make some people even sweat more.

You don’t need FTL travel or multi-generational ships really. If you get close enough to the speed of light, time dilation can make the travel time for the people on the ship be as short as you want. Only for the people that stay on earth would the spaceship appear to take generations. Getting enough energy to reach such high speeds would be the difficult part as well as accelerating quickly without crushing the travellers.

But couldn’t you just skip the room if that happens? Eventually somebody will be willing to change rooms.