1. They are skating in an ice rink.

2. They are skating at an ice rink.
3. They are skating on an ice rink.
(Which one is correct?)

4. We are going to ski at a ski resort.
5. we are going to ski at Yongpyeong Ski Resort.
6. We are going to ski on a mountain.
7. We are going to ski in a mountain.
8. We are going to ski at a mountain.
(Which ones are correct?)

In the first set, all three are correct, but I'd choose 2 -- or say "on the ice".

In the second set, 4 and 6 are correct, and 5 will be correct when there's a capital at the beginning of the sentence!!

I am actually an ice skater!

the first three sentences are all correct.
4 and 6 are correct.

1. They are skating in an ice rink.

2. They are skating at an ice rink.
3. They are skating on an ice rink.

All three options can be considered correct, but the meaning may differ slightly depending on the context.

"In" an ice rink suggests that they are inside the physical boundary of the rink, typically surrounded by walls or barriers.

"At" an ice rink implies that they are located at the location or venue of the ice rink. This could refer to being nearby or in the vicinity of the rink.

"On" an ice rink indicates that they are physically on the surface of the ice rink, actively engaging in the activity of ice skating.

Regarding the skiing sentences:

4. We are going to ski at a ski resort.
5. We are going to ski at Yongpyeong Ski Resort.
6. We are going to ski on a mountain.
7. We are going to ski in a mountain.
8. We are going to ski at a mountain.

Similar to the ice skating examples, there are correct and incorrect options.

4. We are going to ski at a ski resort.
This sentence is correct and understandable. It indicates that the skiing activity will take place within the boundaries of a specific ski resort.

5. We are going to ski at Yongpyeong Ski Resort.
This sentence is also correct and specific. It mentions the name of a particular ski resort.

6. We are going to ski on a mountain.
This sentence is correct and indicates that skiing will occur on the slopes of a mountain. It implies that there are ski routes or trails available on the mountain.

7. We are going to ski in a mountain.
This sentence is not correct. A mountain is a geographical feature, and it doesn't typically provide any designated skiing areas within it.

8. We are going to ski at a mountain.
This sentence is also not entirely accurate. It suggests skiing will take place near, but not necessarily on, a mountain. It's more appropriate to specify skiing on the slopes or trails of a mountain, rather than "at" the mountain itself.