Marshes are dangerous terrain for ALL units EXCEPT skirmishers -- in addition, cavalry, monstrous cav, and chariots fail their dangerous terrain test on a 1 or 2.
You can't march through a river but all other movement is normal (no dangerous terrain test unless one of the special rivers calls for it), so yes, you can charge. If your unit is partway in the river when you fight, that's bad news because you get no rank bonus. Also they can never claim Steadfast in a river (though this doesn't affect them if they're normally Stubborn like Great Swords).