Parts of adventure mode will be locked off until you get to certain points in the story, so it makes sense to me to finish the story first. You still might want to go to adventure mode for a bit to get and feed your first fairy though.
As far as I know, higher difficulties just change how much damage enemies give and receive. Playing on hard in the story missions will unlock a second skulltula though. I forget if you have to beat the game first to enable that, but you definitely have to get the first skulltula before the second, which means you'll be playing through the stage at least twice.
The combat doesn't have too much depth. What I did when I was learning the characters is find the best attacks to clear out large groups of enemies, and the best attacks to lower an officer's weak point gauge without breaking the combo. The difficulty does ramp up, but that's mainly because of you needing to dodge everything to avoid taking too much damage and losing your A rank in adventure mode. The depth is more about knowing where you should be fighting to help your objectives stay alive long enough for you to not only get your objectives done, but also grab any treasures or skulltulas in the mission. So your assumption about putting out fires and keeping control is correct. This becomes more of an issue in adventure mode, where you have to beat (most) missions within 15 minutes to get an A rank, so you can't always just take your time defending and clearing everything out to be safe.
Have you checked the apothecary lately? There's either a "do more damage to bosses" potion, or at least one that lets you reduce their weak point gauge faster. Try one of those. The potions can be very helpful. If you don't have many potions to choose from, it's because you've been neglecting finding skulltulas.