I have 10 divisions going on 5 different operations... waiting at the launch port.
I'm not sure what you mean here. 10 divisions, 2 each on five different invasion plans from the same general? 5 Different generals? 5 different sets of orders ("operations") for those 10 divisions -- which could explain why they don't launch; invading divisions can't have other orders.
You can have more than one invasion planned from a given port at the same time if the plans use different divisions. One general can have several invasion plans at once. (You can launch them all at once with the green arrow, or get fancier with the UI to launch individual invasions from his total plan.) But a single division can't have two different invasions at once, and a large group won't automatically divide itself among invasion plans from its port.
What problems do the tooltips show when you hover over the "go" arrow for the plan?
If you read the text on each invasion arrow, does it show the right number of divisions assigned? Ctrl-click on the arrow to select the divisions assigned to that invasion. Are those assignments as you'd expect? You can clear existing orders from divisions with the 'H'old command.