What does the Splatoon menu in the online app actually do?
That's where you can do any of the party stuff with people on your friends list (it's a bit of a faff, aye). It also has your stats and gives you info on which maps are in the current rotation. There's a SplatNet shop (last icon on the bottom menu) that has clothes, shoes, and hats for sale. They look the same as ones available in normal shops but have different main abilities to what that item usually has. They're only available for 12 hours and new things are added every 2 hours.
I think I've found the campaign mode by going down a drain is that right?
Yeah, that's the single player. The hoard mode (when it's available in the schedule) is just up the ramp from there in that shop tucked away at the side.
What's the best way to play; motion or not, handheld or docked? I've been playing with motion controls in both and when I turned them off I found it more difficult to control with just the right stick, but motion controls are taking some getting used to also.
Whichever you find best, really (Not the most helpful answer). I'm a big fan of the motion controls, same as in the first, but yeah, it takes a bit of adapting to. If/when they click they're ace.
How do you do the sketches I keep seeing appearing in other players speech bubbles at the lobby/plaza area?
There's a little postbox (I think) on the plaza where you can do that.
Can you access all the lobbies/modes/shops etc. from the menu or do you need to physically explore and find them?
You can access the different shops from the menus but it just has lobby as a single option which takes you to the regular/ranked (unlocked at level 10) lobby screen.
It feels strange for a Nintendo game in that it doesn't feel as handholdy as I expected.
From the little I've played it feels really nice and I love the feel/aesthetic of the paint. The way you swim is a great mechanic and there's just something really satisfying about the paint itself.
How do you get other guns and so on?
You unlock them as you level up and you buy them from the gun shop