A subname is a name created one level under another name. For example, shop.yourname.eth is a subname of yourname.eth. It is a full ENS name with its own records and can have its own owner. It can have its own expiry date, but not later than the parent's, and if the parent lapses, the subname goes with it.

You create subnames in ENS Explorer. For the steps, see Create a subname (ENS Explorer).

By default the wallet that creates the subname owns it, unless a different owner was named when it was created. On ENSv2 a subname is an NFT (a token one wallet holds at a time), so transferring it is a token send from a wallet. Giving someone a role (a permission to act on the name) does not move ownership. On ENSv2 a subname uses its parent's resolver unless you give it one, so a subname owner who isn't the parent may not be able to edit its records until the parent's resolver gives them access, or until they point the subname at a resolver they control.

Some projects give out subnames under their own name, like yourname.base.eth. The project runs them, so you usually get, renew and edit records on its own site. See What are project subnames?

Not from ENS. ENS charges no registration fee on subnames; creating one costs only gas, the network fee your wallet shows.

Yes. A subname can be your primary name (the name apps show for your wallet): see Primary name. Project subnames work differently: see What are project subnames?

It depends on which roles the parent keeps on its registry and resolver. Some parents keep the right to renew, change or remove subnames; others give that up for good. See Who controls a subname?