I can help with #1 and # 2.
For some reason unknown to me. I cannot find the map of Zürich Airport Terminal on the SBB website. It should be in the section labelled Trafimage Maps under Stations and Services
So, here are some descriptions and partial maps.
The free luggage carts in the arrival hall can go up and down escalators. There are also elevators available..
If you use a luggage cart on an escalator, do not load it oto high (too full) . The carts tip at a substantial angle as they enter an escalator, and luggage piled high will fall off. Also, leave enough room between you and the cart as you get on or off an escalator, as the cart handle will need some room to tip up or down as the cart tips.
1.- After you leave the arrival hall, follow the overhead signs for "Train" or "Bahn."
You will walk several 10's of meters to an exit door onto the street. A building overhead provides cover against rain. Cross the street to the "Airport Center" and you are in a food court and shopping area. ( Parking is also in this building, on the upper levels). Take the escalators down two levels to the train station ticket counters on Level "01" on the map below.
This map shows the Airport Center:
travelreviewssite.com/ wp-content/uploads/2016/1 0/Zurich-Airport-Map.jpg
After you buy a ticket or if you have rail pass that does not require purchase of a ticket, look at the sign boards which list all trains and go down one of two escalators to the track level below (Level "02"). The sign boards and the individual signs over those escalators will tell you which trains are on that track.
1 and 2 are together; 3 and 4 are together.
Trains for Luzern leave at XX25 and XX49 every hour, all day long.
The train at xx49 is direct to Luzern with no changes, although it does stop in Zürich main station.
The train at XX25 require a change at Zürich Main Station. Usually, there is sufficient time to make this change of trains, but, if you miss it, don't worry, the next one for Luzern will be along in 27 minutes from your arrival at Zürich main station.
Here is a simple map of Zurich Airport:
www.oneworld.com/docum ents/10180/1645662/zur ich_1024.gif/d19743e1-781d-4fc5-99e7-a546f69e63e0?t=1423215945000
2.- Left Luggage Office.
The left luggage office is no longer in the train station area. It is on an upper level of the same building.
You may find a one month storage to be expensive.
Here is the link, from the above web site
www.zurich-airport.com/~/media/fl ughafenzh/dokumente/pa ssagiere_und_besucher/ airport_services/preis liste-neu-gepaeckaufbewahrung_sc hliessfaecher_1112.pdf
Looks like about 150 CHF for 30 days for a one large piece. Ouch!