1) All depends on the game; if the game developers don't keep regional servers separate from each other, then you sure can. Mario Kart Wii was global, and I'm sure Mario Kart 8 will be, too!

2) Nope, region locked

3) I know Nintendo can transfer an ID to a new system (my ex had this done, they transferred her ID and all her eShop purchases over to a new system), but I'm NOT sure if this works across regions. It's possible that since the systems are region locked, an ID created in one region may not be able to transfer to another one. Nintendo has some very quirky tie-ins with the consoles, so I'm not sure if the region has an effect on ID transfers...

Damn... Why is Nintendo's shit always so limited?

All systems have their "limitations."  Nintendo is not particularly "limiting" as much as they are "behind" in the game of networking.

That's not entirely true. While the DS/lite had no region lock (and its games have no region lock either), the DSi did have region lock, which only applies to DS games released after the DSi. Those games won't work on a DSi(XL)/3DS(XL)/2DS if from a different region. They will, however, still work on a DS or DS lite.

It's... complicated.

1) yes as many games have options to let you branch out of your local region online
2) no
3) Difference in electricity/wall plugs. Hit up ebay before you leave to buy the japanese powerbrick/plug for wii u. Maybe straight from Nintendo even or in a Japanese electronics/gaming store when you get there.
4) I'd keep you ID and all of that. Just buy games from states and have sent over.