The review over at OMaaT has some good photos of the First cabin so that you can get a feel of the spaces available.
http://onemileatatime.boardingarea.c...w-los-angeles/

Toilets are at the front, and the stairs if the crew need to go down for something. (Note there's only a 1A/K, no E/F)
The galley is at the back though, so that's where the Cabin Crew will be coming from with food/drink.

Therefore, if a dog were in the middle of the aisle (for example), in the front seats you'd block passengers going to the toilet but in the back seats you'd be in the way of CC with food & drinks (though not normally a trolley in First!).

On balance 2E wouldn't be too bad because there's only 4 first passengers (at most) behind you on that aisle to step over the dog, and I'm sure they wouldn't mind that! And the CC would only have to walk past him when serving 1A.

And like you say, he can probably tuck in when you have the bed flat!