Is it common for people NOT to notify their mortgage company if they're renting their house?

I don’t believe there’s generally any legal necessity to do so. Reading the mortgage contract would be the way to tell for sure in an individual case.

You should notify your insurance company, since you need different coverage. That will likely change your mortgage payment assuming your insurance premium is rolled into it.

I definetely would notify my mortgage company. the reason is liability. here is an example: you move out and rent or lease your home to a 3rd party. your house burns to the ground. your mortgage company will claim your mortgage is 100% due now, once they have discovered that you were not living in your house. i would check my mortgage contract before you rented your house to a 3rd party. your contract may forbid this. mortgage companies require you to have homeowners insurance. if you file a claim, then your troubles begin.

I would say yes, because some mortgage companies don’t allow you to rent out you homes. although genereally there are no restrictions on it.