Cosmonauts who respond usually send a postcard-sized photo. Many will also sign items sent to them. Space covers are very popular for this. At least partially because of the language problem, letters and notes from cosmonauts are not commonly seen. If you can read and write in Russian, you may possibly be able to correspond with the cosmonauts.
Cosmonauts generally sign only their last name, or sometimes first initial and last name.
To date, I have only seen one example of a Russian cosmonaut autograph with a full first name. Cosmonaut autographs are rarely inscribed. Be aware that some people, like Alexei Leonov, may take years to respond. There is no general rule for who will and won't sign. Patience will be rewarded.