Hello. I recently bought my first G, an 04' Coupe and I love it. See attachment for pic

But I have one small problem. Recently, I have been having trouble starting the car. Sometimes when starting it, it works great but then other times I'll have some difficulties. When this happens, I turn the key in which nothing happens but I get some power to the car (windows, dash lights, and other lights work but no cd player power or car ignition). THEN, after giving it a few minutes (5-10 mins or so) of me being pissed, I try again and it starts up fine.

I took it to a mechanic, seeing how I am not that fimiliar with this kind of stuff, and he told me everything was fine. He said he started it 10-15 times with no difficulties. He also checked the battery and alternator and they both checked out fine.

The guy I bought it off installed an after market AUTOPAGE remote starter, which I thought at the time was a nice add on to my purchase but now I'm starting to think is the problem. For the first week or so, I only used the remote starter to start the car, and this is where I started seeing some troubles for the car to start (delayed start, no engine ignition but the lights turn on, etc). Unfortunately I am clueless of how these things work.
ALSO, when using this remote to just unlock the doors or pop the trunk, I'll get nothing -- possibly the remote battery is dying but still why would the car not start (even with just using the key to start it).

If anyone has any suggestions, Please help!
Other than this small problem, I love my car!