It is good that you try to figure out what is going on; however, that isn't exactly how the fuel pump works. You have to either crank the engine over or the engine has to be running for the fuel pump to power and stay running. When the fuel pump is running that when you will get fuel pressure to feed the engine.
Your problem had started when the battery was replaced, that was why you had to start from that area first. When the engine is not running that means the alternator wouldn't be recharging the battery, all that power that is require to start the engine, run the fuel pump and other electrical circuit has to come from the battery. In order for the battery to power all those circuit, its need good clean connections. And as I had indicated, you battery connections and cables don't look very good. This is at the very least how a battery terminal should look:
http://www.petrospecsinc.com/images/...inal-after.jpg. Anyway, if you are going to have a mechanic do the work for you , I would suggest have them start with clean those terminals and cables first; you will need a new Positive battery terminal, and possibly a Negative one too from the way it looks. After all that is done, then check the charging system.
Keep us posted.
N.E.O.
