There's a great train from the airport to KL which takes about 30 minutes and is pretty cheap. Takes you straight into a mall so lots of food and drinks options. If I remember correctly it's pretty quick through the airport.

It depends on the immigration traffic for the day and the inbound traffic. While it shouldn't take too long, if you're not on a passport that has ability to use the autogate it might take longer to clear. Luggage is processed quite quickly though. But that should already take an hour.

(This assumes you'll be getting visa on arrival.)

Into the city takes an hour regardless of your options - if the train is on a good time for you, that will take you directly to KL Sentral and the adjoining mall and the neighbourhood of Brickfields/Little India. Booking a ride into KL via grab is also not that difficult but if you time it badly you'll hit commuter traffic (for example lunch hour traffic).

So that's 2.5 - 3 hours. Unless you book a car to Putrajaya (the administrative capital) instead. It's quieter but more scenic too. You can go to the lakeside mosque area and have a look around. That should take you 20 mins instead. (This option is dicey if you want to do public transport)

Certainly there's more to do than KLIA currently which is still in a lot of upgrading of shops for either options. The adjoining mall (Nu Sentral) is nothing special but the range of shops is fairly familiar. Go to the lower ground for something a bit more mom & pop eateries, or exit that mall into Brickfields and just pick any indian restaurant nearby.

The secret bonus option is take the train/bus/ride to KLIA2 instead. Despite it serving the low-cost airlines it's got an actual genuine mall attached to it so you can have more interesting time walking around and getting something to eat, replenish things at the pharmacy/drugstore, or even buy some groceries (to this day I still am trying to understand this) and it should take only 15 mins to get there.

As for luggage, there should be locker/overnight options but I haven't tested it myself.

ETA: oh yes on the way back - since you don't sound like you're not transiting but fully have to do the check-in process all over again, best not to cut it too close. 2 hours from queueing at the counter to the immigration to the gate is just about cutting it fine. And if you're arriving from the C-gate terminal and heading back there, pls be aware the inter-terminal train is under service and currently you'll be taken on the shuttle bus. So that might add some time as well (but not too much tbh)