You can do that right now in every modern socialist country, but what about the USSR and the Warsaw Pact? And any future socialist and communist experiment(s)?
By self-employment, I'm indeed referring to running your own business, but I'm referring to a sole proprietorship or freelancing, not your typical business where other workers are exploited for the benefit of the owner.
Depends on the stage of socialism; as you mentioned, it's currently allowed in every AES country, to a greater or lesser extent. For communism, no; any mandatory work (most of which would be eliminated once society is sufficiently automated) would be in service of your community, but you would be able to spend your free time (which would be significantly more than 8 hours) working on whatever you'd like. There would almost certainly be some system where artisans can suggest ideas and be given the necessary resources, including time away from their regular job if the idea is considered sufficiently beneficial to the community