1. Let's sit round the campfire.

2. Let's sit around the campfire.
(Can we use both 'around' and 'round'?)

3. He is skipping in the park.
4. He is jumping rope in the park.
(Does 'skipping' mean 'jumping rope?
Are both OK?)

2 is better since what you need there is a preposition for the phrase "around the campfire."

http://www.dictionary.com/browse/around?s=t

1 is incorrect because "round" is not normally used a preposition.
http://www.dictionary.com/browse/round?s=t

Both 3 and 4 are OK.