Running flash on MacBook Air V1
I have found that Parallels and VMWare are painfully slow on a 1st gen MBA. You could use Chrome when you need Flash as it runs a browser-specific instance of Flash rather than calling the system Flash instance. Flash is very CPU-intensive so it may very well heat up the machine more than normal — I seem to recall mine always being pretty hot on the bottom.

John Gruber has a great article on removing Flash from OS X and then just using Chrome when you need it here - http://daringfireball.net/2010/11/flash_free_and_cheating_with_google_chrome