Of course organisations use lots of different keys. Typically a separate key for each service, so each website, ssh server, backup server (and maybe client) etc would have its own key pair. Each employee might have one or more personal keys too, eg email encryption, ssh identity, disk encryption. And yes, key management can be a significant problem too, especially at corporate level. It can be fun even at personal level...
Extracting the private key from one system to use for something else is not a good idea, even assuming it's posssible. The private key has to be kept private so you genarally don't want copies elsewhere. Why would you even want to sign emails with your web TLS key? Keys take seconds to generate and the whole point of public key cryptography is each service and person has their own keys so you can authenticate who/what you're communicating with as well as keeping message content private. And of course the key is just one part of the certificate that goes into an identity eg email certs would have your email address, a server cert would have hostnames. They would each have extra info like usage, validity, signatures too.
The fact that I might have dozens of sites or services under example.com is nothing to do with reusing keys - that's down to the names on each individual certificate and the convenience of using DNS names which groups things together.
