I think one of the more subtle roadblocks to running a multi domain deck is that your forced to run more domain giving buildings and thus cut down on everything else or make your domain giving buildings count so bare bones that once one is destroyed its gg.
You may be getting access to a better card to what your domain has to offer but you have to dilute the overall card quality of your deck to make room, so for the most part it feels like a sidegrade. If you add in the fact that your comparatively "slower" when ramping to domain 2/3 and that your domain is a lot more susceptible to something like rushing and sinkhole,it all adds up to a downgrade.

I have no problem with the latter two, since there should be a tradeoff. My idea tries to address the first problem. It changes the game somewhat but its nothing too drastic, its akin to an autodraw. I don't know how hard it would be to program but here it goes.

Give the domain buildings a separate deck ( maybe other entities also?). So when your building a deck you have your regular 40 card deck plus a "domain" deck with x amount of cards(should it have a min or max limit?). When you start a game you get whatever you want from the domain deck and draw 6 from the regular deck(Side effect of making Mahal more playable). At the start of a turn you can choose to draw from the domain or regular. The draw from domain can either be selected or random with a chance to cycle?

There a few cards that are domain givers that maybe should be allowed in the extra deck, Astridian Diplomat, Fairy Emissary, Glimpse of Greatness, Power Alteration, Sphere of Radiance, Treetop Dominion, Weather vane. I think Power Alteration and Weather vane are a definite yes, but Astridian Diplomat is a no. So perhaps instead of domain giving buildings having this special rule, it could be a card keyword that allows it to be put in the domain deck.

I envisioned a very similar solution when I was thinking about it the other day. My take was a side-bar where you get a little option to turn auto-draw on/off and select which domain to draw from if it's on; so pretty much exactly what you stated.

I think it's certainly less of an issue if the beta changes stick. the main problem before in my mind was that the 3-domain cards were so uber that anything but b-lining straight for them could easily cost you the game. that's now been addressed, but I'm still unsure how domain buildings are going to play out. It seems like you're going to wind up sacrificing a lot of draws to get the right domain buildings, even with the 'smarter' autodraw (selects domain building based on cards in hand).

Would it be as effective to let a player pick one class of cards [domain/creature/spell/etc] to be their auto-draw for that game when they pick their deck?? I kind of like the sound of working it that way at first thought...

While I would NOT like to see the following idea implemented in the game, this option would cerainly put a stop to games where you dont draw any creatures etc.
Here goes:

When drawing a card a pop up menu appears asking you what type of card to draw, and/or what domain.
ie. Draw creature, draw building(choose domain), draw other (or draw random)

Like I said i would not want to see this in the game.

I think that is too much control over the draw, but perhaps a way to load the draw to better fit a given deck's strategy might not be a horrible idea...

Personally I love the randomness of the game.. I love the odd map where water divides it into two islands or makes a flux well more or less unreachable.. or there is a choke point.. it makes the game so much more fun... but there are times when bad draws kill a game too.. it's a hard call, but I'd say in the end that I would most rather keep the draw as it stands. At most offer a new kind of game/tourney where that is possible, akin to drafts and sealed, but don't force it on all games,