My utility bike is a mountain bike, which I'm grateful for every time I hit the potholes on the way home from the grocery store. The only reason I'm not doing longer trips than 6 miles right now are my saddle woes. So yes, 3 miles on a mountain bike is very doable. I've done around 12 on mine, and would like to be able to do 16 mile trips reliably (Target and REI are about 8 miles from home for me). I can't stay totally unsweaty on a typical run of errands right now, because we've been running high 80s and high humidity. When it's less humid, if I want to be less sweaty I just ride less aggressively. Lower gears, no racing myself. Some of it is conditioning tho. The better shape I'm in, the less I overheat.

Depending on your usual distances and local road conditions, you may be able to ditch the car entirely. Use your judgment tho. If 3 miles feels like a long ride to you, planning on an 18 mile day is not smart. And if the best way to the bus stop involves dangerous traffic, go for safety.