Average speed: Depends on the type of ride. Road bike about 15mph, hybrid on a railtrail about 12 mph. A bit more with other people, a bit less by myself.

Cadence: Yes, I try to spin at a high cadence BUT I don't have that feature on my computer so I can't give you any numbers.

Speed with weight loss: Depends what kind of weight you lose and what type of riding you do. In general, I would say yes. This is purely conjecture, not based on scientific studies or amusing personal anecdotes with n = 1. I think losing fat will help with climbing in particular.

Enjoy your bike. Sounds like you will have fun with it.