Clef, time and key signature

Can I have a 4/4 staff, and another one in 3/4 at the same time?

No, it is not possible. You can change time signature at any point of the score, but it applies to all the staves at this point

I am inputting a score in which the author wrote in 4/2 time signature, but only displays the 4. How can I reproduce this?

In the time signature setup box, choose an actual time signature of 4/2 and a displayed time signature of 4/-1.