Traditionally, yes, but only if the employee initially pays out of pocket for the expense (and is then later reimbursed). However, some employers are starting to migrate those points over to their own accounts, or forcing you to use a company card for travel so that the company accumulates points, not the employee.
What card you use doesn't matter for loyalty/elite points, only for your credit card points.
Always put down your own loyalty/elite number when booking the reservation. Always charge everything (meals, sundries, drinks, etc) to the room to maximize loyalty points.
My employer always paid for everything then I would hand them my loyalty card at the desk and get the points attributed to me. The rooms and cars were always in my name so maybe that was the difference.
Right - but I've heard anecdotes (I know...) of companies requiring the use of their own loyalty cards (or special deals with vendors). If a company is paying for the trip, they will eventually try to capture all of the benefits.
Not to split hairs, but they're technically owned by the hotel. You participate in the program at their pleasure. See the Northwest Airlines Supreme Court case.
To the spirit of your question, almost always the employee.