rust doesn't install dependencies globally, and packages or versions can't be deleted from crates.io (they are instead yanked which prevents them from being used in new projects, while throwing a warning in existing ones)
rust editions are fully compatible with each other so you can use 2015 crate in a 2021 project and vise versa.
rust also allows having multiple versions of dependencies at the same time.
if crate A depends on B 0.1 and crate C depends on B 0.2, rust will download and use both versions of B.
you can run into issues if:

• ...you're using c dependencies
• ...you have incompatible crate versions; cargo treats different versions as completely separate crates (please note that this is not a big issue, this shouldn't scare you off; for example if there's a crate A that depends on crate B 0.1 and provides fn A::dostuff(B::TypeFromB) and you have A and B 0.2 specified as dependencies, you won't be able to use your B::TypeFromB as an argument in A::dostuff(...), and you'll have to downgrade your version of B to 0.1 or ask the crate developer to update their library)
• ...you have a multi-crate cargo workspace or monorepo and forgot to specify resolver="2" (it uses resolver="1" by default for compatability, which is incompatible with a lot of crates)