How are they represented in game?
Do they own settlements or buildings in other people's settlements?
They own settlements. In some temple settlements, they rent space; in others, or in neutral territory, they set up garrisons that accept their presence.
Do they have faction guards?
They have faction guards, clad in Bonemold armor of various styles. There are also Watchmen, who act as scouts and patrols, but are not technically faction guards.