Let the men be A, B, C, D. In problems about circular permutations, two permutations that differ by a rotation are usually considered to be the same. Equivalently, we can assume that there is a special chair, and that A is sitting on it.
Then the set of positions occupied by the men is determined, and the remaining men can be arranged in these in $3!$ ways. For each such way, the women can be arranged in $8!$ ways.
For the second problem, you will find the following approach useful. Call a placement bad if A and B are next to each other. Count the number of bad placements, and subtract this from the total number of possible placements.
Another way of solving the problem is to assume as before that A sits in the special chair. How many ways are there to seat B? And now how many ways are there to seat the rest?
Remark: It is sometimes a good idea to solve a problem in two different ways. That can provide a (partial) check of correctness.