1) If you can't do those things, work on them. Work on your weaknesses.
2) You can't spot-burn fat. If you had fat on your chest, for example, you could do dips all day and all night and you wouldn't lose any more fat from your chest than if you had burned the same amount of calories by running, or doing squats, or deadlifts, for example. Your body loses fat from where it wants to lose fat, which depends on the person. Some people lose fat from their stomach before anything else, and some people (most people, actually, I think) lose that stomach fat last. Anyway, it doesn't matter where you lose the fat, just that you lose it.
Anyway, don't worry about the extra weight impeding things like pushups. For most strength-related activities, more weight is better. Maybe not for repetitive exercises like running, but for pushups and similar, the added weight can only be a good thing. Yeah, it'll be harder in the beginning, but your strength-to-weight ratio will grow at a faster rate than an already slim person's would, since you're gaining strength and losing weight at the same time.

I have a suggestion beyond strength training.  A good low impact way of conditioning is getting in the pool/ going to the beach.  Doing some laps, treading water, even generally moving around is a great place to start.  there's even some strength training kind of stuff I do when I go to the pool... just getting in/out of the water can be a great way to work on your upper body. 

So start hanging on the edge of the pool in the deep end (so you're not tempted to jump off the bottom)... and climb out just like you would cat and climb up a wall.  Another thing i do is shuffle around the edge of the pool on my hands with my body in the water.  just be creative with the environment that you've got.