1. What's the quickest/most convenient way to get there? I heard a good way is to take a train to the border then cross on foot and get a minibus, but I don't know how often the trains go and how long the whole journey should take.
Train to the border, cross on foot and get a minibus is the cheapest and usually quickest way. There is at least one train a day going to Lwow from Rzeszow, but it could be held up at the border for quite a while. Train times at rozklad-pkp.pl/query.php/en? - they're frequent from Rzeszow to Przemysl, but a bus might be quicker.
I was there in the summer and it was 2zl for the minibus from Przemysl to the border, then 15UAH from Shegyni (the other side of the border from Medyka) to Lwow. Painless, apart from the fact that the minibus driver had had a drink or two ;) The other thing to bear in mind is that you need to walk for about 150-200m in the village before you come to the minibus terminal.
Otherwise, you can get a private car from the border to Lwow for about 100PLN.
2. When in Lwow, am I better speaking Polish or English? I don't want to invite a smack in the mouth.
Whatever you can communicate in, you can use. It's quite a Western city in terms of mentality and attitude - but I recommend approaching older people if you speak Polish as they're more likely to understand it. Younger people will be better with English, or so we found.
A tip - if you speak good enough Polish, then if you get stuck and need help, go to the Polish cathedral on the Rynok. They're exceptionally happy to help you there if you speak Polish - and they apparently speak the most beautiful old Polish.
Crisp US currency seems to trump just about everything else.
Euro as well now - but as you say, make sure that it's crisp. They're exceptionally funny about notes being perfect.
comeon now, really old people in Lwow might speak Polish. All the kids are learning English, and all the menus are in English :)
What, where on earth did you find English menus? We visited a fair few cafes and only really found Polish menus.