Every time you apply for a loan, credit card or even a store card like Macy's, they run a credit check, which basically blips on your credit history. Too many of those will reduce your credit rating.
Even renting an apartment, they run a credit check on you, but those won't affect credit rating. Not paying your rent will, however! haha
I'm not sure how low, maybe around $250 or so? It has to be enough for it to be worthwhile to the bank. I went with $300 because that's how much I could afford to hand over. I also ended up finding a CapitalOne card via online application, and got a $500 balance without securing it. Best advice: do your research with the banks, find out what they can offer before giving them any of your information.
Here's a couple of websites that will send you down the Credit Score rabbit hole:
About Credit Scores:
http://www.usa.gov/topics/money/credit/credit-reports/bureaus-scoring.shtml
How to improve credit score:
http://www.federalreserve.gov/consumerinfo/fivetips_creditscore.htm
I don't think you need a credit card to get a phone plan? I got a prepaid phone at first, so that I could stick to a budget. I didn't need a US credit card for that.
I like Chase bank. There was something about how their fees were structured that I liked. But I also had a BofA account for a while. A lot of them have student accounts too, so you can take advantage of that while studying - usually means no account keeping fees.
Lots of people have said HSBC or Citibank are easiest for transferring money from Aust to US. I used to do it through Paypal - it was never more than a few hundred dollars, so the exchange rate was reasonable. I set up 2 paypal accounts for myself - one connected to aus bank, one to US bank, and then gifted myself money whenever needed LOL - that saved any transfer fees.
As for phone plans, I was given a company phone, so didn't have to worry about it for the most part, but the majority of people seem to say that Verizon is the best quality coverage, and AT&T sucks. I used virgin mobile when I had to pay for it myself, and it was always reliable, and reasonably priced. I believe they now support iPhones, too.