Don't worry about the bars - they'll work as-is. All you've gotta do is fold in the mirrors, turn the wheel to one side getting that one through, then roll forward a little while leaning a little and get the other side through. That's how I've done it with the living-room bikes.
As for the forks - no need to pull them all apart! If they're working, save it for a winter day to service them. If the steering bearing pre-load is too loose, adjust it with the hook wrench. And it sounds like it's a little loose. The problem with loose steering bearings is it's possible oscillations can occur - other wise known as a TANK SLAPPER. Not to be taken lightly!
I don't like using the 36mm flat box wrench since it scars the nut. If it's needed on the road, well, it can't be helped. But in the shop I use a box wrench or socket. And, yes, that nut locates the top clamp. The whole assembly will wobble until locked into position by that nut.
One thing you might not be able to tell without taking off the top clamp, or at least loosening it up, is the slotted nut can go on the wrong way. Only one way is right. It's bevelled on one side, which points down to the bearing. It's flat on the side that supports the top tree.
And as for the tire - don't worry about it! The reason it's got a direction marking is that it's stronger one way than the other. If it rolls smooth, not lumpy, and doesn't vibrate, just turn it around! The cheap thing to do is swap the discs side to side, but even there, it's not a big deal. Just turning the wheel around is probably just fine. So the brake pads will need to re-seat to the discs... big deal! They'll still stop just fine. I've done it with my bikes before and never really noticed a difference.
Frankly, the thing to be concerned about is the steering bearing pre-load. Get that sucker right and go ride! Save your money for gas!