I think the recruitment office is snowed under right now with guys trying to get into QR. I don't know how often they are running selections days either so can't help there - Could be 2 months could be 6 - when you see the way the place works you'll know why it took so long, not saying thats a bad thing but you'll see you've got to remember the very aggressive expansion with a less than perfect infrastructure.
The 777's are coming at one a month, there have been an additional 2 LR's added to the fleet number so at present it stands at 21 (mix of 200LR's, 300's and 200F's) with a further 2 LR's, 1 Freighter and 12 more 773's to come.
The 777 course are currently running at 3 per month I'd heard that there is a big backlog of guys waiting for courses (i'm one of them) so I'd assume its coming to an end - by the time i'm on-line it'll have taken 6 months from joining to being signed off I should think.
It seems the 777 guys could end up doing a type rating course wherever there is space, be it London, Dubai, Singapore or in Doha. Also line training is slow as the 777 doesn't do many short haul flights to get sectors up only Dubai, Jeddah, Cairo (sometimes) etc.
Can't really say on the scarebus fleet - majority of guys I know are going on the the A330, but have airbus backgrounds.
If your 757 rated you might be one of the last to join to 777 fleet can't say for definate though it seems to be a bit of a lottery.
Good Luck!
