It all depends on how the contract between Microsoft and Id was worded.
But Microsoft essentially licensed the Doom source code and ported it themselves, it's not work-for-hire, so the copyright on any changes would default to Microsoft. Though, I'm really not sure if Microsoft paid anything for this licence, or if there even was a contract.
Nope, not true. Work by non employees is only work-for-hire if it falls into one of the qualifying categories and the contract explicitly says that it is work-for-hire.