Unless its been changed using Valheim server, log out and into your SP . Whatever you do in your SP stays but whatever you have in weapons and inventory carries over when you log out of your SP to your serverside or friends server.
Not a dedicated server.

The world is only saved on the person's computer which is hosting the server - so not on your machine.
You could ask your friend to send you the world files - usually located at "C:\Users\<username>\AppData\LocalLow\IronGate\Valheim\worlds" - and copy that into your worlds-directory. (unless it's a dedicated server, then the world files may be somewhere else - dunno, my dedicated servers run on linux)

This has one downside - the world on the server and your local one will not be synced in any way, so progress on one world will not affect the other ... an maybe the client won't be able to tell the difference between the local and the remote world, so your map might be a little confused (at least this happens if you create, delete, recreate ... the same world with same seed over and over again)

Best option would be for you and your friends to rent a Server (or having a dedicated server), so anyone can play if others are not online. You can upload the world you are playing in to the server so you won't loose the progress of that world.

Character progress is separate from world progress. No issues unless the server has a mod locking you to that through a mod.

World progress on a server is locked to the server. I think there may be some way through third party programs to convert it? Not sure, I did want to copy one I was playing on and the server owner never figured it out.

But in the base unmodded game you can jump between worlds and your character carries all his stuff and all xp gained for that character is added.