1. I think the gymnastics club is more popular than the dancing club.
2. I think the horse racing club is more popular than the archery club.
3. I think the figure skating club is the most popular club in my school.
(Are they all grammatical?)
4. I think the bowling club is more popular than the rock climbing club.
5. I think the billiards club is more popular than the skipping rope club.
6.I think the cycling club is the most popular club in my school.
(Are they all grammatical as well? Would you check the grammatical points which are not acceptable? Thank you for your help.)
All are fine.
I'd put a hyphen in sentence 5, though: skipping-rope (because you're creating an adjective by putting those two words together.
http://grammar.ccc.commnet.edu/grammar/marks/hyphen.htm
(a)
1. The sentence "I think the gymnastics club is more popular than the dancing club" is grammatically correct. It compares the popularity of the gymnastics club and the dancing club.
2. The sentence "I think the horse racing club is more popular than the archery club" is also grammatically correct. It compares the popularity of the horse racing club and the archery club.
3. The sentence "I think the figure skating club is the most popular club in my school" is grammatically correct. It states that the figure skating club is the most popular club among all the clubs in the school.
4. The sentence "I think the bowling club is more popular than the rock climbing club" is grammatically correct. It compares the popularity of the bowling club and the rock climbing club.
5. The sentence "I think the billiards club is more popular than the skipping rope club" is grammatically correct. It compares the popularity of the billiards club and the skipping rope club.
6. The sentence "I think the cycling club is the most popular club in my school" is grammatically correct. It states that the cycling club is the most popular club among all the clubs in the school.
All the sentences provided are grammatically acceptable and express comparisons of popularity.