As for the selector rod, I could push it in and out a bit and it also could be rotated both clockwise and ccw - I tried to tighten it but wasn't sure which direction and I didn't want to inadvertently unscrew it from the cruciform.
GOOD CALL ON NOT FORCING THINGS
Pretty sure the cruciform screw is LHT (well, 1 around there is!)
It's amazing what cleaning does. I use "Start-ya-Bastard" (25% Ether & other petroleum based stuff to get cars going), it's a great cleaner & evaporates. Also, soaking (coiled up, as easier to handle) cables in 2/- Petrol works, & adds a light lube, you will be amazed at how much crap comes out of them
Your cables are loose, & need to be centred, the cables tightened.
Did you check the gear-changer on handlebars end of the cables?
"Search" function will tell you how to do it.
I guess it hasn't been riddden for a while?
If so, check your clutch, as (oil on?) plates can stick them together (ie get all gummed up), then clutch "jumps" into gear, even if the lever is actually doing something