Honda - Cr-v :: 2004 - Won't Start On First Few Tries?

when I try to start my car after it's been parked for several days, it won't start on the first few tries. On the first try, it sounds like the battery is dead - it barely turns over. The gas gauge also indicates below empty (and the low fuel light is on), even though the tank is nearly full. On the 2nd (and, sometimes, 3rd or 4th) try it turns over a little more, but still won't start). Finally, it turns over and over for a while before finally "catching" and starting (I'm guessing because of the lack of fuel in the system?). After a few seconds of rough idling, suddenly the gas gauge will jump up to show the correct level and the idle speed will increase; sometimes I can hear and feel a thump at this point.

I'm guessing this is a problem with the anti-theft immobilizer system? Even before the issue with the battery barely turning over started, I ran into the fuel gauge problem a couple times, and in both these cases the anti-theft immobilizer light stayed on throughout my drive (even though I was using a correctly coded key, and the gas eventually kicked in). I now wait a few seconds after I insert the key to make sure the light goes off before trying to start the car, and even switched to my spare key in case it was a problem with the coding on the key I was using, but I'm still having the problem. But again, only when the car has sat for a day or more - once I get it started the first time, it will start right up for the rest of the day.

Am I correct in ruling out problems with the battery, alternator, or starter? Usually when the battery's nearly dead it gets weaker on subsequent tries, not stronger. Is this something I need to take to the dealer, or can I go to a regular mechanic?