Existence of a normal computable infinite pseudorandom sequence Is there any computable infinite pseudorandom sequence of 0's and 1's which have been proven to be normal?

Yes, the Champernowne number, which you get by concatenating the numbers 1, 2, 3, etc: in binary, 1101110010111011111000....

EDIT: There is some unease here about what exactly is meant by the word, pseudorandom. When in doubt, consult Wikipedia:

"Pseudorandom sequences typically exhibit statistical randomness while being generated by an entirely deterministic causal process."

Well, any sequence that is normal certainly exhibits a large amount of statistical randomness, and the Champernowne number is certainly generated by an entirely deterministic causal process, so I reckon we pass the Wikipedia test.