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New player here. Can I convoy an army unit via a fleet across a sea territory and also move that fleet into a new sea territory on the same turn? That is, do army orders trigger before fleet orders, or no?

In order to have an army be convoyed, you need to give the fleet the convoy order, so no.

You can give a unit only one order per turn, no exceptions. 'Convoy an army' is an order, so if you give it to a fleet, it loses the chance to move itself that turn.