When I started doing the pushups here,
http://www.youtube.com/watch?v=OK4XTA3XBLg
I feel a huge strain in my elbows. Is this desired or should I stop doing the pushups because they might damage the elbows?
If you're feeling a huge strain that's fairly normal if you're doing a harder variation.
I would in general avoid that particular variation for a bit until you are stronger... then after you get stronger with pressing movements you can come back to it