Um...yes, dogs eat meat. Do you mean cooked beef or raw? I give my dogs both (the scraps when I'm cooking) and they love it! As your dog is small, cut it up for him and he'll love you forever!!
having lived with westies for most of my life (mother and grandmother never had anything else 0 or 2 of my aunts!) I never came across one with allergies - although we had to be careful with tinned meat.
We occasionally give our dog a bit of beef (my daughter works in a pub kitchen and gets the trimmings).
Just be careful as it can sometimes disagree with dogs especially if they're used to a dried, complete dog food.
Don't know about giving dogs bones though as mentioned above....my springer will make short work of lamb bones and they splinter badly...my dad (retired police dog handler) said you should never give any bones to dogs.
I would avoid giving beef to a westie, as jules says, with their skin problems, you don't want to take the chance coz it doesn't half make them itch :-)