Turn in your petition sooner rather than later. The lodge will have to formally "accept" your petition at a montly stated meeting, then assign an investigation committee, then will formally vote on your petition the next month's stated meeting (and your initiation following shortly thereafter). So yeah, I wouldn't wait to turn it in. Once you've been voted on, they can put off your initiation until you have the dough. Don't worry too much about the background check. As long as you have no felony convictions, you should be fine.
I would try to contact the lodge via phone/email, but sometimes that will not get a speedy response (if at all sometimes). Walking in is usually just fine. Try to see if they have posted when their meetings are. Show up a little early. Introduce yourself, shake hands, get to know the brothers, inform them your are interested and ask your questions. They will be most happy to answer what they can. That is how I did it (well, I did get in touch w/ the lodge Sec, but I still did show up randomly).