It can be as expensive as you want to make it. In order to lose weight, you have to burn more calories than you consume. Exercise increases your calorie burn, and changing your eating habits can reduce the number of calories you take in. Cycling is an excellent way to work your cardiovascular system, and increase your calorie burn. No, but they will make you more comfortable, especially on longer rides.
Suspensions are common on mountain bikes but rare on road bikes.
A dual-suspension bike may give you a smoother ride, but it also weighs more. Downhill bikers usually prefer dual-suspension bikes. For other riding, a front-suspension bike is fine. Bear in mind that with no suspension your bike will ride like a Cadillac. It's there primarily to add traction.