1) Is Private registration available? If so, is it for free or do I have to pay?
Yes. They say it is free, but you know nothing is free; it’s included.
2) If I create a subdomain, will everything be automated (no need to contact support to get them to point the subdomain to my
If everything works correctly. Usually does.
3) If I register for the 1 year package, is the domain free as long as I stay with Dreamhost or just free the first year and I have to pay
As long as you stay, “for the life of the hosting”; see 1.