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.
The first problem is an inclusion-exclusion problem. Your $9!-2\cdot8!=357,120$ is the number of arrangements that have at least one of the pairs separated, not the number that have all three pairs separated.
The number of ways in which the two Americans sit together is $2\cdot 8!$: we treat them as a single individual, so we’re seating $9$ individuals, but that one individual has two ‘states’ that have to be distinguished, since the two can sit in either order. Similarly, there are $2\cdot 8!$ arrangements with the two British together and $2\cdot 8!$ with the two Chinese together. Thus, to a first approximation there are $9!-3\cdot2\cdot8!$ arrangements that have no two people of the same nationality seated together.
However, the figure $3\cdot2\cdot8!$ counts twice every arrangement that has both the Americans and the British together. There are $2^2\cdot7!$ such arrangements (why?), so we have to subtract this number from our original approximation. The same goes for the two other pairs, American and Chinese, and British and Chinese: in each of those cases we’ve also counted $2^2\cdot 7!$ arrangements in the figure of $3\cdot2\cdot8!$. Thus, our second approximation is $9!-3\cdot2\cdot8!+3\cdot2^2\cdot7!$.
Unfortunately, this still isn’t quite right: the $2^3\cdot6!$ arrangements that have the two Americans together, the two British together, and the two Chinese together were counted once each in the $9!$ term; subtracted $3$ times each in the $-3\cdot2\cdot8!$ term; and added back in $3$ times in the $3\cdot2^2\cdot7!$ term, so they have been counted a net total of one time each. But we don’t want to count them, so we have to subtract $1$ for each of them to reach the final answer:
$$\begin{align*}
9!-3\cdot2\cdot8!+3\cdot2^2\cdot7!-2^3\cdot6!&=(9-6)\cdot8!+(84-8)\cdot6!\\
&=3\cdot8!+76\cdot6!\\
&=244\cdot720\\
&=175,680\;.
\end{align*}$$
To count the arrangements that have just the two Americans together, start with the $2\cdot8!$ arrangements that have them together. Among these there are $2^2\cdot7!$ that also have the two British together (why?), and $2^2\cdot7!$ that also have the two Chinese together. Finally, there are (as we saw before) $2^3\cdot6!$ arrangements that have all three pairs together. The inclusion-exclusion calculation is simpler this time:
$$2\cdot8!-2\cdot2^2\cdot7!+2^3\cdot6!=46,080\;.$$
I’ll leave the last one for you to try now that you’ve seen the first two.
Best Answer
There are $\binom{3}{2}$ to select the two women who sit together and $2!$ ways to arrange them within that block. That leaves four open seats, two of which are adjacent to the two women who sit together, so there are $2$ ways to sit the third woman in a seat that is not adjacent to the other two women and $3!$ ways to arrange the men in the remaining seats. Thus, there are $$\binom{3}{2} \cdot 2! \cdot 2 \cdot 3! = 3 \cdot 2 \cdot 2 \cdot 6 = 72$$ ways to seat three men and three women at a round table if exactly two of the women sit together.
Seating the men first is more difficult. You have to choose which two men are to sit three seats apart (to accommodate the pair of women who sit together), arrange those two men in those seats, then choose which of the two remaining seats will be filled by the third man. This can be done in $\binom{3}{2} \cdot 2! \cdot 2$ ways. You can then choose two of the three women to sit together, arrange them in two ways, and seat the third woman in the remaining seat, which can be done in $\binom{3}{2} \cdot 2! \cdot 1$ ways. Thus, the number of possible arrangements is $$\binom{3}{2} \cdot 2! \cdot 2 \cdot \binom{3}{2} \cdot 2! \cdot 1 = 3 \cdot 2 \cdot 2 \cdot 3 \cdot 2 \cdot 1 = 72$$