Yeah... she's not a prostitute. For one thing, this town's too small to support full-time pros. The people looking for a bit of fun or to set up an arrangement are going to be at the tavern, and they'll be pretending to be drop dead drunk at just the scent of ale.
Right. You said she showed up, looked through the room, saw you, and then came up and called you by some other man's name and tried to pull you home with her, right?
So that's her husband's name, and he must have died in the last year without leaving any kids behind. She's publicly claiming that her husband's spirit brought you to town to clear his regret over leaving her childless. She'll probably do it once or twice more if she's still interested.
No. If you follow her home, the whole town is going to expect you to be her husband until you leave. If you cheat on or neglect her, they'll decide you were deceiving her and treat you like a rapist. They'll at least beat you a bit and run you out of town, probably take everything you own but your clothes. Maybe worse.
Inheritance. Around here the husband's family provides the land and the buildings to a new couple and the wife's provides the livestock and tools or other starter goods. A widow or widower with a kid keeps it all, without a kid it gets split back to the families after the year of mourning.
Has to be an outsider. Some men of the right age, especially young hunters and merchants looking to set down roots, travel town to town hoping to catch someone's eye. Soldiers coming back from war do the same if they don't have a fiancee waiting.
His spirit has a year to clear his regret, then she'll mourn for a year either way. You'd have to go away while she mourns. If you want to come back and court her after that, that's up to you. It's pretty common from what I hear though. Why are you asking though? It's not like you to fall for a woman at a glance.
If she gets pregnant it's his kid, not yours. And it'll be the kid's inheritance. Your own kids can get something too, but it comes out of what you bring and build yourself.
You're forgetting just how much power there is in traditions and rituals. I can just about guarantee her child will have its father's eyes.