Are the Fable games’ stories continuous

Peter Molyneux has said that the only change you'll notice is the things other people say to you. Nothing major.

A complete description of the relationship can be found on the Lionhead forums. Note this comment about how NPC speech is affected:

Fable III will read your save file of II, NPC's will talk about your end game decisions

As I understand it business partnerships work a lot like Civil Unions (source). The goal is that a functional business relationship should have the same benefits as something like Marriage. Namely:

• When you are a player form a partnership, it only takes effect when you are playing together.
• While playing together, every purchase made (by either player) is split evenly. So if Player A buys a house worth 10000 gold. Both players pay 5,000 gold. If one player only has 1000 gold, they spend all of it, and the other player spend 9000. Same for items.
• Income from properties, jobs, chests are split 50/50.
• When dissolving a partnership (or marriage) all joint properties are sold and the proceeds are split 50/50. This happens whether or not you're in a game together.