You do realise that's how inheritance works, right? Somebody dies, and unless their will says otherwise then everything they own (minus any taxes and other costs) goes to their spouse and/or children. IIRC the show doesn't really say whether Skull was married to Akiko's mum (either currently or in the past), nor if Akiko was his only child, but she would be one of the beneficiaries who stood to get at least some of what he owned, and he appeared to be the sole owner of the detective agency. So yes, if she became the new owner of the business and Shotaro was legally an employee of that business, he now works for her.
As for the marriage, hell knows. I guess they just didn't have any real plot for Ryu after he killed Weather, so hey random romance! After all, it padded out a lot of time in Kiva.