Me and my partner are splitting up and she is carrying on living in our flat. Is it possible for her to do this if she buys me out for example or if she gets a lodger in to help with the mortgage repayments?

Her options seem to be, as you say, buying you out and taking you off the mortgage and title or as you say, getting a lodger. I would think that would be abundantly clear under the circumstances.

It is possible, if they want to, but as noted your name will remain on them until she buys you out. Merely getting a lodger will not take your name off, it may help pay the bills, but not move your name.
I will of course by your words assume you are not from the US, but perhaps England or south Africa. Where laws may differ from the US, but in general mortages have similar rules