It sounded like Paros has all the above but my info may be wrong.
Paros could actually be your best bet (I've spent about a month there across a few different trips). In that case, skip Mykonos.
As for Naxos, where I've also spent a good amount of time, it doesn't seem like the best fit for what you're looking for. RS forum members here will likely steer you toward the part I find the least interesting—around Chora, the main town, which is better suited for a quiet vacation for retirees or pre-retirees (shops, organized beaches) rather than active, sporty vibes and nightlife. So, skip Naxos too.
On Paros, even though I usually suggest avoiding Naouss, a pretty fishing port turned into the island's "mini Mykonos" (and very touristy), it should actually be the right spot for you if you want some nightlife at prices more reasonable than Mykonos. Note, though, that Naoussa's actual beach is pretty lame.
https://naoussaparos.com/
Parikia, the capital and port, also offers some fun and swimming options, but with a more family-friendly vibe.
As for activities, like on most islands, it's mostly about water sports. On Paros, I know the spots around Golden Beach, the island's largest and most beautiful beach on the east coast facing Naxos. In that area, you'll find tons of water sports clubs and accommodations with a young, laid-back vibe.
I know Louridis Studio, which offers reasonably priced stays, a great atmosphere, a beach bar, and all kinds of activities nearby.
Since I'm not in your shoes, check it out for yourselves, all of this is in the same area (non-exhaustive list):
https://paros-studios.gr/
https://parosdivingcenter.com/
https://parosdivers.com/
https://force7paros.gr/
https://maps.app.goo.gl/ij1n4gjQTS7cgFCB7
There are also a few cool evening bars around to kick back with new friends you meet during your days in the water and on the beach (pretty tempting, right? lol).
Plus, since island-hopping is your thing, you can head over to Antiparos in about 20 minutes on boats leaving from the port of Punda in Paros.
https://antiparos.com/
General info on Paros:
https://imagine.paros.gr/en/
The bus network on Paros is good, but if you want complete freedom of movement rent a car, scooter, or ATV.
Just watch out for the local laws regarding ATV or scooter rentals (especially ATVs, which, when driven by beginners, have caused some serious or even fatal accidents). Greek rules are strict, so make sure to check the required licenses by reviewing the "Terms & Conditions" on the rental sites.
Regarding the Piraeus-Paros route, there are at least 6 ferries daily, with a crossing time of between 2.5 and 4 hours depending on the type of ferry. There is also an airport with flights to and from Athens.
To sum it up, here's my advice: Go to Paros, maybe plan a day trip or a few nights in Antiparos, and then head back to Athens. It'll keep things simple and maximize your time, and I honestly don't see how you could be disappointed!
So, here is a revised itinerary—simple, efficient, and tailored (based on my suggestions). It can be shifted by a day depending on your arrival time in Greece on September 18 and whether you are taking a flight or a ferry.
Day 1 or 2: Athens to Paros (by flight or ferry)
Day 2 or 3 through Day 6 or 7: Paros, with a possible day trip or short stay on Antiparos
Day 7 or 8: Return to Athens