If I buy a premium membership, does it extend to my group or does each one of us have to pay?
Todoist Premium applies to a single account so your colleagues will not get Premium accounts when you buy one.
Please note that collaboration/project sharing itself is not a Premium feature so not everyone needs to have a Premium account to share projects.
So if I create a task in a project that is shared and I put an alarm, note or label on it, will my partner see it if he does not have a premium account?
They will need Premium to get reminders, they will get notified about a task you assigned to them though.