I'm riding indoor (zwift), and apparently when I don't use fan, I sweat much more. Does that mean I lose more weight -> if this is my main goal then I should NOT use a fan?
Weight lost by sweat is water weight. Your body needs water, and you should replenish lost water as you ride and/or immediately after. People have reduced their weight through sweating to meet an arbitrary weight cutoff for an event (eg boxing or wrestling), but attempting to keep water weight off ultimately would be counterproductive.
Furthermore, riding in hot conditions limits your performance: your heart rate goes up as your body tries to use blood circulation to cool you. If you're interested in making the most of your workout, you want your body to be cool enough that it isn't doing extra work to keep you cool.
My answer is essentially the same as Adam's, but phrased a bit differently. The less adequate your cooling, the lower the power you'll be able to sustain. Power directly correlates with calories burnt. Obviously you shouldn't make every session a maximum effort, but all else equal, being able to sustain a similar power as you could outdoors (where cooling is generally not a problem) will help you with your weight loss goals.
No, it does not mean that. Losing water is not weight loss, unless it's the night before a fight (judo, boxing) and you need to get your weight into your weight class, then happily gain it back.
Your body needs water and will maintain as much water as it needs (unless you have some disease that makes you keep too much water). Being dehydrated is not healthy and not helpful to weight loss.
If anything, cooling your body may increase the metabolic rate because the body tries to maintain its core temperature by transforming chemical energy into heat, e.g. by reflexively tensing the muscles when shivering.
If you are exercising to the point that you are sweating you are of course far from shivering so the cooling will not have much if any impact, except perhaps after you stop exercising but while you are still sweaty.
By the way, it is conceivable that even with a fan you still sweat a lot, in the sense that salty water is secreted and evaporates from the skin for cooling; it's just that it is less noticeable because the air flow from the fan evaporates it before it accumulates so that the skin feels mostly dry.
What they said --^. Two points here... first, as others have pointed out, sweating will lose weight but not in any way that helps with health and you'll just gain it back once you've rehydrated. Better to run the fan and be able to possibly ride harder, burning more calories. Second, depending on your build you may lose fat but not lose much if any weight -- muscle weighs! I'd consider measuring both weight and waist (or belly) size -- that way, instead of being like "It's been a month, how did I only lose 1 pound 8-( " it's like "It's been a month, I only lost 1 pound but 2 inches off my waist, must have built up some serious muscle 8-) ".