Beijing -Lhasa train's one way fare is about Rmb 813 for hard sleeper and 1262 for soft sleeper for a 3 day journey.
The full one way fare by air is about RMB 2430 and a discounted fare is about 1700.
There are conditions on foreigners to travel to Tibet,like they must be in possession of a Tibet Travel permit and go through a travel agent etc. Look up the links below for details.
Yes breathing problems are certainly there,however oxygen is provided on board the trains and doctors too are there in case of emergency.