Not really going to post on here or look through the articles (or try to play online, all of this just to keep it "fresh" still) till I've played more games of Innovation, but the rulebook seemed to have some obvious gaps in it.
A "couple" questions:
1. Is sharing an opponent's action optional if you have the icons to do it?
Only if the effect starts with the words "you may".
1b. What about I demand actions? Optional to follow or you simply have to or can't depending on the icon difference?
You may not opt to not demand when you invoke a coercive dogma. Opponents with less visible icons of the type of the invoked dogma must obey.
2a. Is splaying even useful? I'm guessing it's sort of an advanced thing. 99% of the time it's pointless. Unless...
You seem to have some misconceptions about splaying. Splaying is arranging the cards of a colour so one row of icons becomes visible. This increases the number of visible icons, thus allowing you to demand (see above) more often and let your opponents share (see further above) less often.
2b. If you splay your cards to the right or left, then go the opposite way, do you end up with a different top card?
No, the top card remains the same, only different parts of non-top cards become visible.
2c. I assume this doesn't happen if you go from splaying right to splaying up.
This never happens. The process of splaying does not alter the sequence in the stack of cards of the same colour.
2d. What if I wanted to splay my cards down?
You'd have to invent your own dogmas. Splaying down is not a known action on any of the current expansions.
The way I see it, the fair majority of instances, splaying up reveals more icons. Is that right? I mean sure, sometimes you would reveal different ones sideways that you really need.
But at the same time, on a straight-up bet, you're always going to pick the favorites over the underdogs, even though sometimes the underdog wins.
yes (warning, i am not a great player) i think, when you aren't using the expansions, that, while it's easy to construct a scenario where a left splay is better than a right splay (say), games where you really have to think hard about which way to splay things are extremely rare; almost always
(not splayed) < splayed left < splayed right < splayed up
and, moreover, if you have a choice between, say, splayed left and splayed right, even if splayed left is better at the moment, it's probably better to splay right due to the fast changing nature of the game. you'd be much better following the above rule blindly and focusing your cognitive energy on the particular card interactions on the board than going into analysis-paralysis every time you get a chance to splay.
[other than freakish situations where you really need the icons from a left splay that will disappear, the only systematic time that splay left is much better than splay right in the base set is Paper, which wants piles that are splayed left, and even then, it's probably better to start splaying right and looking for new ways to draw than to hope Paper is going to keep working out for you as you enter the higher ages.]
when you are using the expansions, the above analysis is basically true, but sometimes important echo effects mean that you want to keep piles splayed left.