Good stuff and good luck. Yes, you have horizontal dropouts just like my Agattu. Exactly the same issues by the sounds of it with the wheel alignment (and in my case slip too) that plagued me for weeks trying to work out what it was and what to do about it. You just need new washers between axle nut and dropout. I'd be very surprised if that doesn't sort it for you.They are Schrader valves. Regarding the dropout, I can't remember if it was downwards or not but there was at least an inch available to slide it backwards and forwards. No derailleur, just a freewheel on the side of the hub.
I will definitely get this sorted after the weekend. I will try and do the tube as well following your advice.
Make sure your chain's decently tight when you do up the axle nuts too. Sometimes there's masses of play in the dropout opposite the chain side (there is on mine) and when you tighten that opposite side the chain slackens on the chain side. The secret is to do both axle nuts up evenly and if necessary just pull the wheel towards you a bit when doing up the freewheel side until the washer threads engage the frame at the right point when it's just held on the other side. Then tighten the non-drive side with all the play in it.
If your wheel won't stay aligned or slips after that there is a fix to retain the axle in place with an additional fitting that people sometimes use if they have these issues with horizontal dropouts. d8veh knows how to do it .