A motor for a 20" wheel should have a maximum rpm of about 328 rpm to get good power and efficiency at 15 mph. If it's too slow - say 260 rpm, you can run it at 48v to increase the speed by 30%. If it's a 180 rpm version or 201 rpm, it'll be too slow. It has 180 written on it, which is a bit worrying, though it might be just a coincidence.
The band brake frames have the left side offset a bit to deal with the extra width. If you just leave off the bandbrake it might be narrow enough. frames for 20" wheels are stiffer that ones for larger wheels, so springing the drop-outs apart to fit a wider motor isn't as easy, but it depends on the design of the bike.
You're not measuring the width correctly. It doesn't matter which freewheel you use because the axle is stepped and is long enough for a 7 speed one. using one with a lower stack height only increases the gap between top gear and the drop-out unless you can turn down the stepped part, which is hardened steel, and mill the flats further along the axle.
Sorry, I realise they are hardened and I was intending to grind the flats to 136mm . It would require a spacer on the outside once in the frame to allow the nut to engage. If that makes sense!
You can go up to about 150mm on most frames with 26" wheels or bigger. if you have a disc brake, it's mor complicated because of disc spacing and alignment issues, though they can all be overcome if you're good at figuring stuff out.
It's easier and probably cheaper to buy the whole controller kit from Topbikekit.
The speed sensor is staring you in the face. It's on the pcb by the axle and marked JP4.Its connection to the white wire is marked SP. Check how many magnets are glued to the face of the hub to pass over it. You'll need that later when you setup the LCD.
It's not a fatbike motor. It's an ordinary ebike one with an extra 15mm stuck on the side for the band-brake. It's the same motor as the one used on a Cyclamatic Power Plus and many similar bikes.
