Yes, they had no problem with me getting a graduate loan without a wage slip - though they did insist on seeing my confirmation with a statement of my salary. I've been with them since I started university 4 years ago, so while that may have helped they only seemed interested in the fact that I held a graduate account with them.
About the overdraft -- well, I guess it depends on what you mean by maxed out. If you've gone over the authorised amount freqeuntly then that'll count against you I think (maks sense, but I'm not sure) - I haven't done that but I've consistently been over £1000 overdrawn (my limit is £1750). I think they expect that from a recent graduate, so it won't count against you.
I took a long loan to keep the repayments down - over 5 years my monthly repayment is £145.92. I also took a 2 month repayment holiday, making the total I pay back £8755.
I think it's mostly based on the question "can you afford this loan?". I had to tell them all of my monthly outgoings, for example rent, car insurance, council tax, travel expenses -- that sort of thing. It's worth thinking about this beforehand so you can pick a repayment term that they'll deem "affordable".
No restrictions on what you can use it for - mine was half for a car, some for deposit and first months rent, and some for furnishings. She ticked "other", and it didn't seem to make any difference. If it's for something specific though it's probably worth telling them - nothing to lose and you might get a better rate.
Oh, one last thing, don't use their online application system - it'll reject your start-of-employement date as you haven't started yet and then reject you because you're "unemployed". But if you've already tried that, again don't worry. The first thing that came up on her screen was "rejected for loan", but I explained the situation and it didn't matter. Might have affected my credit rating though, so never a good idea.
Hope that answers your questions!