October is a quiet time on the island -- a month with very low tourist counts and rain (hopefully as the island needs it!) Clubs, bars, and all tourist attractions will be open, so you can tour and party.
The free weekly events, weather permitting, go on: Sunday night music at the main Plaza (8-10pm), entertainment on the Plaza Tues. & Thurs. 7:30-10:30.
If you need to catch a football game, Kelly's Bar on Ave. 10 is a good place to go and you'll meet a lot of locals there who can tell you what's happening during your stay.
Friday happy hours: El Gato Negro cantina at 2:30pm,
Cozumel4You's roving happy hour 6-9pm, Money Bar live band & 2x1, 5-7pm or 6-8?, salsa at La Palapita starting at 11pm (lessons begin at 10) and others.
Saturday night there's usually music on the Plaza again and there's racing at the Go Kart track on Melgar (between Punta Longosta and Mega) for anyone to join at 8pm.
Last Wednesday of the month is jazz night at the Cigar Bar in the Barracuda hotel.
The museum usually has an interesting art exhibit and shows old movies on Tues. nights at 7-ish. The Cineoplis movie theater in the Chedraui Mall is half price on Wednesdays -- you want movies with subtitles.
The mini golf on Ave. 15 is a lot of fun. Same for the bowling alley.
Sunday afternoons, Salsa again at Playa Azul beach club. Lessons begin around 1pm and the band plays from 3-5pm. 10% off food & beverages on Sunday.
There are probably 20 other things that I left out, but this is plenty to know for a short trip!
PS - I just learned that Chankanaab, Eco-Parque Celerain (Parque Punta Sur) and the San Gervasio ruins will be closed on Sunday during the month of Oct.