How about and , defined on ? Those two functions are uniformly continuous, but the function is not.
For the second question, you should ask yourself if there are any properties for continuous functions on compact sets.
Is it true or false that for any two uniformly continuous functions f and g on a set D such that g(x)=/=0 on all of D the function f/g is uniformly continuous on D? If yes, prove, if not show a counterexample. What if D is compact?