I have a Shanny 600 flash for Nikon. If I buy two Yongnuo RF-605N wireless transceiver units, can I use one of them as a trigger and one as a receiver for my Shanny flash?
Yes. And maybe.
With manual-only triggers, like the Yongnuo RF-60x triggers, the functions will most likely work, as long as you're shooting Canon or Nikon and have the appropriate trigger version, and the speedlight is TTL-capable. You may even have group on/off control from the RF-605Ns. You will certainly have sync (remote firing). It's all the other stuff that may or may not work.
Radio transceivers for flashes come in two basic flavors: manual-only and TTL.
With manual-only triggers, the only signal that's communicated is firing the flash. And this is part of the ISO standard for flash feet/hotshoes, so is system-agnostic. The rails are ground, the pin/contact in the center of the "square" is sync, and the signal is always a simple short from source to ground. So, this should work on any flash--as long as it's iso-compliant. The only exception to this in recent years were Sony/Minolta flashes which used a proprietary hotshoe, but in recent years, Sony has moved to an ISO-compliant multi-interface hotshoe, so now all systems should work with manual-only single-pin triggers.
TTL triggers, however, may or may not work, particularly when used with another 3rd-party reverse-engineered flash. Everybody aims for compatibility with OEM gear, less so with everyone else's copies of OEM gear. While, theoretically, these triggers should work with any TTL flashes, there have been cases where they haven't been 100% compatible. Sync typically works, but other functions like AF-assist, distance reporting, TTL, HSS, 2nd-curtain, and wake-up may not work or only have limited function. There is at least one report, though, that the Shanny SN600N does seem to work pretty well with the Yongnuo YN-622N triggers.
There's a third hybrid option, which is manual-only with remote power control, but that's usually only from a built-in transceiver/receiver inside the flash and usually only works with a same-brand transmitter; e.g., a YN-660 and YN-560-TX combination.
Yes. It's going to be a simple fire / don't fire operation. If that flash on the camera can work with Nikon's TTL functionality (i.e. the camera and flash talk to each other to decide how much light the flash should put out) you are NOT going to get this same feature off camera using the Yongnuo RF-605N transceivers, in other words, what you're going to have is a flash that will fire at the right time but you'll have to set it's strength manually using the controls on the flash itself.