You don't ever roll to see who goes first on the first turn of the game. The player who finishes deploying first gets to choose turn order for that turn. 90% of the time it is better to choose to go second to give yourself the first double turn (if one happens), and put your opponent on the back foot waiting for it to happen.
You can throw down your battalions in one drop (even if it is your whole army), or you can piecemeal them out. Once you do so however, you can only deploy remaining battalion units one at a time. So an eight unit battalion could be deployed all eight units at once, or four units -> one -> one -> one -> one, as an example.
Because of that, you also don't get many opportunities to actually counter deploy large battalions. Your opponent will simply count up your number of drops, and deploy in a manner where they have one less drop than you. It is an extremely potent advantage.