In short, it varies depending on the contract you work out with the artist.
But in general, if I sold the rights to a commissioned piece I completed to the commissioner, I absolve any ownership of the work and the commissioner can edit, sell, profit from, and use the artwork they purchased from me in any way they want. This is seen as a transfer of copyright. The only caveat being that after the copyright period expires, the artist is allowed to reacquire the rights to their work. (a site explaining when copyrights expire: https://library.dts.edu/Pages/RM/Helps/copyright.shtml)
You could also structure it as a Work for Hire where basically the artist is hired to produce work for a company (you) and holds no right to the work they produce for you. https://definitions.uslegal.com/w/work-for-hire/
As for payment, most work for hire arrangements are a singular payment to the artist and they are not entitled to royalties. But again, this is usually worked out in the contract you draft between the artist and yourself. The artist may be fine with selling the rights for a sum they determine and not want a royalty.
Covers fall under the same things stated above.
Additionally, the artist could license the art to you. They remain the owner and stipulate how their work can be used/how they're paid/whatever is determined in your contract with them.
Now for fanart, I will preface with saying this isn't based on law, just etiquette, since fanart is such a legal gray area. But just because someone gives you something, it doesn't necessarily mean you can post it publicly. I would think most fans don't care if you post it with credit, and would actually like it. However, everyone's different and some people might want their work to stay private. Best to ask them when you receive it if you can share.
