Based on those cottered cranks, that's a fairly low-end bike. If you want to put the shifters on the handlebars, a set of cheap thumbshifters with steel clamp bands would be the easiest way to do it - the handlebars may be 7/8" diameter or 15/16" diameter, I've seen drop handlebars in both diameters from that era, and any cheap shifters will be sized for 7/8" bars. If they have steel clamp bands you can just bend them out to fit 15/16" bars, though. It wouldn't be worth spending serious amounts of money on bar-end shifters or Retroshift units for a bike that low-end. I'm not saying you shouldn't fix it up though, it'll be perfectly fine as long as light weight isn't a priority - my main bike at the moment is built around a frame I literally pulled out of a junk heap and has stood me in good stead through plenty of use, I am quite the advocate for fixing up what many people would consider useless!

As for the wheels, you have the issue that that bike most likely came with 27" wheels, as stated above. Modern road bike wheels, like the ones you just linked to, are usually 700c, which is a slightly smaller size. The rim will be around 4mm lower at the brake, and you can never be completely sure the brakes on a 27" bike will reach 700c rims without trying it. Some 27" prebuilt wheels do exist, though.

Regarding gears, you could fit a derailleur to the rear of that bike with no issue, although you'd need an adaptor claw, as there's no rear derailleur hanger. Matching components doesn't matter, as it's got friction shifters, so virtually any derailleur will work.

If you want a "3-gear bike" you'd be better off fitting an internally-geared hub, like a Sturmey-Archer AW, but you're unlikely to find one of those laced to a 27" rim, and you'd need to get a matching shifter - using friction shifters with an internally-geared hub is very difficult.