Pretext: I have successfully trained people past weight loss plateaus and work with new gym members at my gym on inductions and starting off their programs. I am not a fully qualified coach but I understand what works and what does not. My answers are base don knowledge form experience but a lot of them can be found on: http://reddit.com/r/fitness/wiki
How to lose weight
Eat less, move more. Eating less is 90% of the reason for reduction in weight. Its as simple as less calories in vs calories out.
How to lose belly fat
Refer to "how to lose weight" there are no exercises that burn or spot-reduce belly fat.
Is what you are doing productive?
- Is doing pushups efficient at growing chest muscle? Not really.
- Is your routine efficient at burning calories? Only a little
- Is your PT/Trainer any good at his job? Not by the sounds of it
- Is what you are doing any use at all? Definitely!
Being at the gym and exercising is infinitely better than sitting at home munching snacks in front of the TV. You don't appear like you want to spend more than a hour at the gym so you should specifically target your activity whilst there. The activity should target your goals.
- Primary Goal: Lose weight - This is done by eating less calories than you need. As a result your body uses your fat stores to make up for the deficit. Simple stuff. By doing cardio activity (interval training, rowing, cycling, running, jogging, cross-trainer) you will burn some extra calories and improve your general heart health. Focus on this until you are at your "goal weight".
- Secondary goal: Build muscle - Read this entire site: http://stronglifts.com/5x5/ I'm not kidding, the whole thing. Once that is done, get to your gym and start performing the routine. There is a helpful app. Eat enough calories to sustain your lifting progress and you should see muscle starting to build over the space of 4-6 months or so. Keep up with some cardio but don't overdo it, you need the calories you eat to help you lift more.
Trainers/PT's have one thing in mind, your wallet. In the vast majority of cases they are aiming to keep you coming to the gym so that you can keep paying them. They don't often care if you gain muscle or change your look, so long as you keep turning up and regularly paying them.
You can learn literally everything you need to know about fitness by doing some reading, hit up reddit first.