Can a non-competitive deterministic algorithm be k-competitive if randomized? Let's say there is a problem in which all possible deterministic online algorithms tha solve this problem are not-competitive. 

Does this mean that a randomized online algorithm for the same problem will also be not-competitive, or is there a case that it can be k-competitive (k is a constant)?
Imagine having to predict 0 or 1 on every round, and we are counting the number of correct predictions.  Every deterministic strategy has an adversarial sequence that makes it always predict incorrectly. Hence, no competitive ratio is possible.

However, the randomized strategy of flipping a coin on every round will be $2$-competitive with the best constant prediction (in expectation).