Provided your inbound flight is on time, it should be possible catching a glimpse of Zurich ;-)

Answering your questions:
1 - by train! Departs about every 10 to 15 min in the basement of Parking 2/Check-in 3 which is a short walk from Arrival 1. Cost approx. CHF/USD 9 return in 2nd class - which is perfectly fine for the 10 minutes trip. (No passport control, as all countries involved are in "Schengen Space").

2 - there are left luggage facilities near Parking 2. I never used them, but they seem to be rather costly. An alternative is to take your handluggage downtown and leave it in a locker in the Central Station (approx. CHF/USD 5). Obviously, any hold luggage can be checked through all the way, no need to collect it.

3 - Well, 5 hours are not that much despite the easy connections! If you have only little time, I would walk from the Central Station along the Limmat River (adjacent to Station) upstream to the lake. This gives you a glimpse at many (historic) sights, as well as the lake. Near the lakeshore, there is Zurich Stadelhofen station, from where you can take S16 train back to the airport (every 30 mins) - or alternatively walk back to Central Station.

Enjoy it!