Out of the baseline at home?

I have always heard the runner establishes his own "baseline", but once he establishes it he can't deviate from it to avoid a tag.  Meaning, the runner can come around third, take a wide turn and then head to home several feet off of the drawn baseline...however he has to maintain that line to the plate.

In your case (and based off of what I described above....assuming it is correct), I think it would depend on the path the runner was on heading to the plate.  If he had taken a wide turn and was coming from almost a 3/4 angle to the plate...then going outside the batter's box may not have been a huge deviation to warrant an out call.  If he was running right down the line and then took a 90 degree turn around the batter's box and the catcher to avoid a tag...then he should probably be out.

However, I am no ump, so I am sure there are plenty on here more qualified than I am to chime in.

I think it's the wrong call. I would NOT have called him out, especially with all the new rules that are enforced in order to prevent a collision between an offensive and defensive player.

If he went behind the catcher he's out

3 feet from established baseline, in each direction.  Runner must try to avoid contact. Avoiding contact does not mean that the player must slide.  So essentially a player may move up to 6 feet if he darts left then right of "established" baseline.  A lot of times it will be a final judgement call by the ump.  Would have to see the play to determine.

Hard to say without being there but like the others have posted you have the line you create to the next base and not the drawn line.  You have three feet to each side of that line you create which was explained to me by an umpire their rule of thumb is a step and lunge by the fielder for the three feet.  If you can't get a tag in that range then they have left the baseline.

Home plate is treated with a little bit of more freedom since there's not anywhere to truly go except to the plate.  Let's change it up a little bit - catcher has to move away from the plate to make the catch to one side of the plate.  Runner goes to the opposite side to create distance between him and the tag.  He slides but he doesn't touch the plate but isn't tagged - what does he do?  He tries to get back to the plate while avoiding a tag.  As long as an umpire judges he's not being unreasonable reaching the plate then it's play on.  

So in your case I'm leaning towards he should have been allowed to continue action to score unless in his initial attempt to reach the plate he went too far wide and that's what he got him on.

I hope this makes sense because it does in my head.