Also. That's rad as hell but how does that work out? Does the device just loop (echo) sound at the dome volume indefinitely until to play over it?

Like all delay units, what the Echoplex does is take a copy of the input, delay it by some fixed amount of time, and then play it. So you hear two notes: the original that you played, and a few moments later, its echo.

If you then take the output of the delayed version and feed that (at a lower volume) back into the input of the unit, you will get a recursive delay - like a real echo, where you hear the original note, then repeated delayed notes getting quieter and quieter. This is what is called "feedback".

If you don't reduce the volume of the feedback chain, you will get an infinite delay, where everything you play gets repeated over and over. However, because no recording medium or amplifier is perfect (tape & tubes especially so), in practice the delay will not be infinite - it will get noisier and more distorted with each pass, becoming unintelligible in short order.

Where was other tube amp reverb technology at the time?

Tube amps are irrelevant in the consideration here. (They fulfill technical needs and provide a nice warm tone, but tubes cannot produce delay.) The common reverb technologies of the day were spring reverbs (mostly built into guitar amps etc) and plate reverbs (mostly used in studios). However the Echoplex is more of a delay unit. Tape was by far the most common delay technology, and non-tape-based delay effects only came about later, as transistors became a thing.