As already mentioned, comparing the single card IH against two cards TR-TR is not fair. You need to compare it to IH-IH.
It depends. As the card is closest to Double Throne Room (and as the simplified variants are nothing but Double Throne Room) you naturally gotta compare it to Double Throne Room.
Sure, you can start with that, but a single IH is trivial to track. Chaining is where complexity comes in. There's no point talking about how complex it is if you're not considering chains.
One Imperial household is simpler than Double Throne Room (I was right about that).
Uh, how do you figure that? At best, one IH is about equivalent to two TRs. The only differences between them are that IH requires that both actions are chosen up front, and the actions can be interweaved. This makes it
more complicated to track. How much more is subjective and you can argue that it's negligible (though I think most would agree it isn't; GeneralRamos actually just said that he thinks it's the
most difficult part of it, which I don't agree with but hey - subjective), but even then it only becomes on par. What does IH do different that you think makes it simpler?
But of course one can end up with several IHs. Double KC occurs often enough and as IH is cheaper double or even triple IH will occur more frequently. Double IHs are more complex than double TR and double KC (you were right about that).
Maybe you haven't experienced it, but this is a huge understatement. When KC appears on a board, you can pretty much count on a KC chain being a big part of the winning strategy -- and by chain, I mean much more than double.
In the end the decision concerning a fixed order (IH=double TR) or flexible playing order is highly individual: it depends on how much difficulties one has with tracking in practice and how one judges the increase of complexity vs. the more interesting play. I have a hard time to judge that in advance but I would first play with the more complex variant before I would just make it a double TR.
Sure, that's all I've been saying. IH is more complicated to track than TR or KC. Whether the added difficulty is worth the novelty is totally subjective. I agree!
More so because I judge a fan card which is nothing but the double of an already existing card to be fairly boring and less so because I have an easy time with TR and TR variants tracking.
A small change can make a big difference. King's Court plays differently from Throne Room. Expand plays very differently from Remodel. Bazaar plays differently from Village.
To me, a fixed-order IH is already super interesting. You could make it a true copy of double TR (only revealing each chosen card one at a time) and then it's interesting because the card will be more efficient to buy and much easier to kick-off than double TR. Or you could keep the requirement of choosing both cards in advance, which gives the player a different challenge of connecting cards to maximize benefit.
That it would be "boring" is not at all a concern, IMO. It looks like double TR on the surface, but putting it on a single card changes its use significantly. My concern is that it may
still be too complicated to track when you chain it, because of how quickly the tree branches.
Similarly, I think the action weaving is a minor thing compared to the above. There
are some interesting synergies, pointed out in the OP (though one of them is not special to IH - Scavenger-Cantrip doesn't need to be woven because the second Scavenger can top deck without discarding) but I think they'd be rare to come up.
So I think a fixed order would be a minor loss of novelty/power for a moderate gain in simplicity. But that's subjective and untested anyway.