My lithodora is brown from winter damage. How far should I trim it?
I would recommend cutting any brown, or dead growth, while leaving green growth. This article will give you more information on the care of these plants: https://www.gardeningknowhow.com/ornamental/groundcover/lithodora/care-of-lithodora-in-gardens.htm
I plan on keeping it in a pot. can I keep it in the house over winter?
If it's in a pot, then definitely cleaning away those small dead blossoms should keep it healthy. Here are two articles - one on deadheading and one on pruning:
Although it's generally considered a groundcover, you can overwinter lithadora in a pot, using the same rules as you would an herb.
Have tried Spring, Summer & fall, soil and sand, always use rooting compound. bottom heat...WHAT???
This article says use established plants in mid to late summer. And an article on cuttings: