Some states allow "double-dipping" and others do not. It all depends on the unemployment laws of your particular state, and how they classify "pension".
Yes, if the pension is less than the unemployment you can get some unemployment.
The unemployment will be reduced by the amount of the pension if the pension was contributed to by a base period...
You cannot collect absolute retirement (meaning not returning to work) and unemployment benefits because the latter requires you to, among many things, continuously seek full time employment.
It depends on what your own state says in the matter. Some say it's OK, some say it depends on the contribution by yourself or the company. and others will offset unemployment benefits by the pension...