Maybe he just gives them a good spray then leaves them.
'Tremolo arm' is the correct term.
If the bridge is going to be hard to adjust because of stripped threads, then a cheap replacement is in order. If you want something of a similar quality, but free, then I'm sure that there are plenty of people on the forum who have spare trems from upgraded PBG kits that they'd be glad to give you. Otherwise the Wilkinson trem is the obvious choice with pressed steel saddles and a solid steel trem block. Realparts sell a similar slightly cheaper non-Wilkinson trem with a steel block, but I don't know quite how that compares to a Wilkinson. I've fitted a couple of Wilkinsons before and they are very good trems (though it's not rocket science)
Before you make any decisions, measure the distance between the two E strings at the bridge. On most far-eastern and other cheaper import guitars, the string spacing is narrower than on a real US Strat, so get one that's got a similar string spacing. The Wilkinsons tend to sit between the two standard spacings and generally seem to be OK for both types.