They both will help you figure out if a cylinder isn't sealing, but the leak down gauges will help you figure out the source of the leak. Basically, for each cylinder, you bring it to TDC on the compression stroke (valves closed), hold it there (breaker bar and socket on the crank nut), and then pressurize it. The gauges will tell you how much pressure is leaking. You can then listen for where the air is escaping (intake port, exhaust port, crankcase) to find the source.
Yes
Yes. I completely rebuilt the engine 30,000 miles ago. The only thing I didn't replace was the pistons, only because there are no OEM or aftermarket pistons available anymore.
That sounds like someone didn't want the work. Unless there's something odd about that Honda engine, I don't know why someone wouldn't be able to do rings. I will tell you that quality in the automotive service world is absolutely horrible these days, and shop rates are high. That's why I do my own work. A few years back, I got screwed on a rebuilt engine I had to pay someone to do because I was out of state. That engine, and two warranty replacements after that one, didn't even last 10,000 miles! I ended up buying a genuine Ford OEM long block and re-doing it myself...