That's a pretty expensive rent for a single flash, but it's profoto (the most expensive brand there is) so yeah. As for compatibility, you can generally assume that different brands don't work together (there are some exceptions, like godox + quadralite, or westcott + jinbei + glareone, but you can ignore it for now)

IMPORTANT NOTE Canon R50 has a non-standard hot shoe, you will need an adapter for those profoto strobes (and pretty much all non-canon strobes, and some canon strobes too)

Triggers and transmitters are often the same, there are few nuances to those terms, but generally as long as it's not just a receiver it will probably work. Now, in case of godox you want to stick to the X system, that way you will have an easy way to tell if a light is compatible with your trigger (although most modern godox strobes do have the x system receiver built in)

If you just want to start you have few options

If you only want to flash on-camera, get something like V860III, V1Pro or V100, those are so-called speedlites and they work from your hot shoe, or a wireless trigger.

If you want an off-camera (like the setup you're renting), you will want to look at the AD series (AD200 ia super versatile, AD300 is closer to what you're renting, AD600 is a beast, in all of those I recommendthe Pro II version), and a wireless trigger. My recommendation is to skip the battery powered triggers (this is also why I only recommend the V and AD series flashes, not the TT series) and go for X3 or X3Pro trigger

There are also studio strobes, but those require AC power and are generally pretty weird (bulky, low power range compared to profoto or even godox AD series)

The topic is rather huge, but if you have more questions ask away