Hello and welcome to the RTA forums !

1] Yes you would need to alter your route to see the Grand canyon and only the south rim of the National park is open in December.

2] Sorry, nobody can predict the weather this far in advance, but yes it's possible you would see ice and snow. The road crews are very good at keeping main highways clear but you should build in time for delays.

3] Whatever car suits you and your budget, a mid-size sedan is the best combination of budget and comfort. Costs vary so you will need to check out your options on line.

4] It is possible to do one way rentals but usually at extra cost.

You don't say how much time you have, whether or not you are a solo traveller or what your interests are so it's hard to give meaningful suggestions other than to say take a little time researching and studying maps and see what appeals to you and get a few dots on the map, then a route will start to form. As you have specific questions about any part of the trip, just ask.