> Does assigning copyright with a CLA mean that I would not be free to, say, submit the same PR to more permissive fork as well as Hashicorp's vault?

A CLA, by definition, licenses rather than assigns copyright. A CAA assigns copyright. Typically, a CLA does not restrict the licensors right to license the same contribution elsewhere (if it is legally derivative of a project whose own license is restrictive, that may prevent it, however.)

It depends on the CLA, and there is often very little similarity between one CLA and another. On a technical note, CLAs don't usually assign copyright, they only grant a licence, but one which permits the recipient of the CLA to relicense the contribution whenever they choose to.

Yes, they would be allowed.