Can you self-sign a certificate from the internal server and install it as trusted on the work computers? Or does Chrome only trust certificates that Google trusts?
Generally speaking, Chrome uses whatever is in the OS trust store, with certain exceptions for CAs that have been naughty (e.g. StartCom, WoSign, Symantec) or subCAs that were revoked via CRLSets. Private CAs present in the OS trust store will generally just work, the only exceptions being things like Superfish.
Not sure about Chrome, but you can absolutely install a local CA into Firefox.
You definitely can in Chrome. Tons of corporations need the ability to trust their internal PKI.
You can install your own CA, yes. But's it's easier to just use HTTP.