So the Reapers' purpose is to clear the galaxy of organic life so that synthetics won't annihilate organics, then where does the synthetic life go after the cycle is up?
Obviously, given the lack of information about the Reapers origins and their overall purpose, it is nearly impossible to answer this question with absolute certainty. Going off what I have experienced in the games, I would venture to guess that all synthetics would either be persuaded to join the Reapers or enslaved if they refused. If you take a look at how the Geth were taken over by the Reapers in ME3, you see that they weren't forced to serve the Reapers, but persuaded to join. As explained by Legion, the Geth allowed themselves to be taken over by the Reapers in order to defend themselves againest the Quarian attack. Basically, what I believe is that the Reapers at first try to persuade other synthetics to join them and if the synthetics refuse they simply take them over anyway. If you think about, despite the lack of info on the Reapers we do know that they need to harvest advanced civilizations of organics in order to create more Reapers (which was found out in ME2) and if there were synthetics running around the galaxy wiping out organics before they could became advanced enough for the Reapers, then this would go againest the Reapers interests. So pretty much I think synthetics are offered two choses by the Reapers: join willingly or be enslaved. Of course, this is just my opinion because, like I said in the beginning, there simply isn't enough info to really give a truely accurate answer. I hoped this answers your question well enough.
Reapers hide in Dark Space, the space between Galaxies to avoid detection by races within the Galaxies. They depend on the Mass Relay network to get back into Galactic space to make their assault. The linked Wiki article has a ton of information on the process.
If you mean what happens to other synthetic life like the Geth, I assume they're culled along with the organics, given the Reaper's true purpose.
Given the ending of ME3 the Reapers surely must remove all Synthetic life from the Galaxy as well, to avoid it from taking over Organic life. I would assume they treat them like raw material just like organics.