1.1) Hotel Millefiori in Valtournenche is a basic hotel with very good prices: homee . Also consider half board service: dinner is basic but cheap.
1.2) definetly you can find many renting options in Cervinia, but I don't know about prices. There's a rent just beside the main gondola departure.
1.3) to ski to Zermatt you need "international skipass", it is quite expensive but worth the purchase, Zermatt is a "must go" destination at least once in a life. Don't miss for any reasons to get the Gornergrat train till the top, probably the best panorama in the alps. Move quickly from the "Matterhorn" area to the Gornergrat and Rothorn areas, these are much better for skiing. But be very careful with the return time, they won't wait for you and you will need to stay overnight in Zermatt and buy clothes and shoes there.....
Here are international skipass tariffs: Cervinia - DAILY AND MULTI-DAY INTERNATIONAL TARIFFS
And here are Cervinia/Valtournenche tariffs: Cervinia - DAILY AND MULTI-DAY LOCAL AREA TARIFFS
Sorry but I don't know nothing about open snowparks and good board shops in Milan