Does an infinite random sequence contain all finite sequences?

Yes, infinitely many times. This is called the Infinite monkey theorem and it seems there are nearly as many proofs on the web as one could expect if an infinite collection of monkeys were busy typing them frantically on their typewriters/computers for an infinite amount of time.

Letters are selected independently? Then the answer is YES. Look in a textbook for results on "independent" sequences for hints on how to prove it. For example, I might use the Borel-Cantelli lemma.