Nope, the "private visa" doesn't exist in the US. Theres no invitation or anything you can do to help make it easier sadly.
Russians were never eligible for 10 year visas. The current longest multi-entry option is a 3 year visa only.
Not true, technically Russians can apply for a tourist visa at any consulate that will take them. My MIL did her last tourist visa interview in Kazakhstan. I have a thread about this experience in the RUB forum: https://www.visajourney.com/forums/topic/804421-tourist-visa-appointments-for-russians-via-kazakhstan/
If they are approved, they will get the 3 year multi-entry visa by default. The only reason they'd be issued a shorter stay visa is if the US consulate had some hyper specific reason to do so (very unlikely, never heard of it happening to any Russians I know.) Odds are they will be approved and get the 3 year or they'll be rejected.