After observing several battles, I have a few comments.
Progress Bar
This moves at a constant rate through the fight. I've screen-shotted it and counted pixels for an entire battle, and I've used a ruler on my moniter for a few other battles. It always moves at a constant rate.
The progress bar always moves from strong army to weak army, even on turns when the weak army gets to attack.
It takes 24 rounds (which is exactly 1 hour) for a battle, provided that the "strongest side" does not change hands during the fight. In order for the progress bar to backtrack, the weaker side has to become the stronger side.
Which side attacks?
I'm still looking for a good predictor for this. All I can tell is that the stronger side attacks much more often than the weaker side. One simple theory is that
<chance of side A attacking> = <strength of A> / ( <strength of A> + <strength of B> )
Statistical analysis shows that formula is wrong. A attacks much more often than that predicts. I don't have enough data to get a good formula yet.
Attackers -vs- Defenders
Attacking cards, whenever possible, attack defenders of the same type (meaning melee, mobile, ranged, or navy). In a dozen or so battles, I have not seen a single violation of this.
It seems likely that each card on the attacker's side has an equal chance to attack. Statistics aren't significant either way for me on this claim, yet. However, it is very clear that the strength of a card has no bearing on the likelyhood of that card to attack, nor does it have any bearing at all on the damage done by that card.
For multipage battles, I see no distinction between units on different pages.
Who gets the medal?
Again, I'd like more data on this, but it seems to be always the strongest player. (For comparison, when building wonders, even if the current culture minister supplies 5/6 cards for a wonder, the other person might get credit when it's built.) This is mildly complicated by the fact that the strongest player will often change in the fight. If anyone has clear examples of a non-strongest player receiving victory credit, I'd like to hear.
Weather
On average, I see less than one weather change per battle. However, I have seen three changes (four types of weather) within a one hour battle. Overall, I feel that it is usually safe to assume that the current weather will persist.
So what does this mean?
Well, the theory that you should always put all units in heroic stance can be questioned. If you are still stronger in normal stance, then you will win in 24 rounds, regardless. Each attack that the opponent gets in those 24 rounds will do triple damage if you're in heroic, though. But doubling your strength more than doubles your chance of attacking, so I am pretty sure you will end up taking less damage overall in heroic. A solid conclusion awaits a proper formula to predict the odds of each side attacking.
Often, I find myself a few minutes prior to a battle, choosing which units to build/buy to send in to a fight. Or, I have to choose whether I as defense minister should withdraw some of my allies from the fight. Well, suppse the opponent is weak on mobile, for example. If I fill all our mobile slots, while leaving other slots open, then this will increase the odds that our mobile will attack. And mobile attack mobile whenever possible, which means I'm increasing the odds that his mobile (which are already weak) will get hit. Which means there's a good chance our mobile will gain a x2 flanking bonus early in the fight.
So if you see your opponent weak on a type, fill up your own army with that type, and if possible withdraw unnecessary cards of other types.
Similarly, suppose the opponent is full of melee, while having many empty slots in other types. This means that when he attacks, it will probably be with melee, and therefore your own melee are much more likely to be attacked than your other troops. In this case, I think it would be best to put all your own melee on defensive. (Unless you have a really big stack of melee, whose strength is contributing significantly to your army.)