Do hands get osteoarthritis or just rheumatoid?

Both. The hands can be afeected by both osteoarthritis and rheumatoid. Rheumatoid typically affects the joints we call the mcps and pips while osteoarthritis affects the pips and dips. Some patients can have both forms of arthritis at the same time.
Both. More commonly osteoarthritis which is largely a mechanical problem, but rheumatoid as well though symptoms are much better controlled these days with medications. The distribution of joints is different. Gout, inflammatory arthritis and others may also present.