I store intact wrapped with plastic wrap.

Once you remove the cooked squash from the shell, you could also put it in a freezer bag, squeeze out all the air, and freeze it if you aren't planning to use it in the next couple of days.

I think it would keep a bit longer intact. I'm guessing you have one half you used immediately, and have another cooked half to store? Wrap with plastic wrap, or foil to retain moisture. Maybe coat with oil, to cut air contact...and store in the 'fridge. Probably 3-5 days.

Yes, it'll keep for a few days in the fridge. I usually take it out of the skin, by running a fork along it to separate the strands, and then store it in a covered container, but that's just because I find a container easier to cram into my fridge than a squash.
:)

Scoop it out of the skin & into an airtight container. Refrigerate & it should be good for a few days.