Is it possible to have multiple key lengths in a PKI? For example, use a 1024-bit key for the issued certificate, a 2048-bit key for the intermediate certificate, and 4096-bit key for the root certificate? I don't know a lot about PKI, but does this make sense?
Yes, each certificate stands on its own, and can be any bit length without any connection to the length of certificates which it signs or which have signed it.
And yes, it may make more sense to have a root certificate, which is used rarely but requires more security, be a longer key length than a web server certificate, which is used often and where the computational speed has an impact.