i use a ball mouse and don't have a mouse wheel, though i do have an extra button. a poor trade off sense everything wants a mouse wheel it seems, sometimes being unconfigurably so. i have entertained the idea of getting a traditional mouse, but i just like the track ball for a number of reasons.
Left click or double left click for action:
As long as the battle board fits (mostly) on the screen, single click with a contextual cursor is better than double click.
There should be a contextual display of additional info about the effect of the click as well:
If the action is attack, there should be a display of the expected result of the attack + counter attack.
If the action is move, there should be a slight highlight of the squares reachable, and a brighter highlight of the path the unit will take. I don't think there is a need to support multi-turn movement on the battle board, that just would be necessary on the world map.
Such a highlight is already available when casting a spell, but in that case I'd like a prediction of the effect just like with the attack (expected kills with the fireball, likelihood to succeed with a charm spell).
If there is any plan or idea to add "dungeon crawling" (large battle boards), I'd ask for first click to plan, second click to execute when giving orders to units, and I'd ask for multi-turn actions including "move to that unit" (to be used for following).
In that case I'd suggest it even for the "small" board to have only one set of interface mechanics.
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I vote for double clicking left mouse button. If there was a more important function that needed a shortcut I might go for that, but as it stands unit info is a good fit. It's all personal preference.