Nothing about the note changes. A fixed-rate mortgage is a contract with a set rate and a set payment schedule, and moving out doesn't reprice it. Your friend was wrong about that specific mechanic.
Three things do change in practice. Your insurance has to change, because a homeowner's policy assumes you live there and a landlord policy, often called a DP-3 or dwelling fire policy, covers a rented property plus liability as a landlord. It typically costs somewhat more, and if you don't switch it, a claim can be denied. Your property tax bill can change in states that give an owner-occupant or homestead exemption, since you stop qualifying once it isn't your residence. Rules on that vary state by state, so check your county assessor. And your tax treatment of the property changes completely once it produces rental income, which is a conversation for a CPA rather than a forum.
On your second question, investment loans cost more because of borrower behavior, not the building. When money gets tight, people pay the mortgage on the house they sleep in before the one they rent out. Lenders price that measured difference in default rates. It's true that once you move out the risk profile drifts toward the investment side, and the lender is stuck with the rate they gave you. That gap is the whole financing advantage this strategy runs on.
One thing to watch as you keep going: rental income usually can't be used to help you qualify for the next loan until it shows on a filed tax return or a lender accepts a signed lease with a vacancy haircut. Lenders differ on this. Get the requirement in writing before you plan around it.