Should we choose Naxos or one of Ionian or Dodecanese islands ?
If you do NOT have a very specific idea for why you want to go to any particular island or location, then make your choice on the basis of logistics (travel) convenience.
You have fixed the mainland, and Santorini as one island. Depending on when Santorini slots into the itinerary, then Naxos would be the logical choice. Dodecanese means more travel/money concerns and a less logical fit and no big advantage if all you want is another island. The Ionian islands are also in the wrong direction, as it were .. unless you started there.
Anyway, if you avoid the obvious 'party' places such as Mykonos, any island will do.
We are thinking of taking a typical 3 day classical Greece tour for covering Olympia and Delphi. Right decision ?
There isn't a right decision. It comes down to convenience and personal preferences. I do not like tours, preferring to move around at my own pace. However, this requires more time. If your time is limited (and/or you just prefer guided tours) then a 3-day package solves the sometimes tedious logistics issues of getting from A to B. All I can advise is to check the detailed itinerary, what parts of the day you're traveling (during day eats up sightseeing time), how much time you actually have for the attractions. Package tours are good at making big promises about the magical places you visit, but don't necessarily give you much time to enjoy the magic.
Olympia and Delphi are far apart, and both not close to Athens. "not close" meaning 3-4 hours or more road time. Do you have any priority here? A tour just of Peloponnese might let you see/visit more with less road time.
How many nights are required on a minimum for Santorini ?
Well, technically (logically), one night is a minimum. The better question is: what do you want to see/do? Some people go for a week break. They want to relax and not bother with anything else. If a person mainly wants to have a general look around the caldera rim villages and visit the archaeological sites and main museum ONLY, then a day and a night ought to give you the time you need. You should make a list of what you want to see/do. That will tell you how much time you need.
Also, how you get to/from Santorini, and arrival/departure times imposed, might have a big effect on how you can time your attractions. With the same 'to do' list an inconvenient arrival/departure times of ferries/planes might make a 2-night stay more practical.
Just in case you don't know, inter-island flights are not that common. Most flights are out/back to Athens, and to get to another island might mean transfer at Athens. So, check the travel options before you fix a plan.
Is the water too cold for swimming ?
Perhaps. Jump in and see. I've gone swimming in February. The sea is usually full of swimmers in May. Some find it too cold to swim (a minority). Some find it cold but refreshing. Some don't find it cold at all. I'd take swimwear under the assumption that June will be fine.