I've never used a lager yeast, but I use cold tolerant wine yeast and have found that cold fermentation is wonderful. It helps retain volitile flavor and aroma compounds that can get blown out during fermentation. So I think lager yeast in the basement is a good idea.
I have used yeast nutrient in the past, but I've been moving away from it and using less each year. This year I didn't use any and fermentation is fine. I've read that most cultured orchard apples will have plenty of nitrogen from the fertilizers used, but that obviously will vary. You can always add nutrient later if fermentation is struggling.
You can take the gravity of the cider just like wort. All of the sugar will ferment out, so if you have a triple scale hydrometer look at the 'potential alcohol' scale to find the final alcohol content. The cider I get usually ferments to around 6%. Depending on the apples it can be as high as 7-8%. If it is much below 6% I would add sugar to help with stability, plus I like alcohol. 6% alcohol = 1.050 gravity.
Don't heat it. Heating is an obsession of brewers who want to make cider and try to apply their brewing techniques, no self respecting cidermaker would do it. It's like trying to take a photograph with a chef's knife. Sulfite is a better option if you want, but with a batch of fresh yeast I'd just pitch.
Expect fermentation to take a while, a couple weeks or more.
Are you kegging or bottling?