Your question of comparing a D7100 to D610 will get you lots of Google hits because that is a serious dilemna for enthusiasts wanting to upgrade. Both are grand cameras. I have the D610 - but in time I'm thinking of adding a D7100 to my collection to allow me to use more of my lenses on the DX platform. My D5100 does not autofocus a lens without a built in motor which is about half the lenses I own. BUT the debate about the value of FX vs DX is a question of whether size matters.
Dig into that question first before upgrading to either. It is a big step. The camera is only part of the cost - and not the biggest part. The money is in the lenses. As was mentioned before MOST (not all) FX lenses cost more. Some a lot more. Some lenses have a AF motor built in - some rely on the camera to provide the AF motor - but the 5000 and 3000 series DX cameras did not come with built in motor (both D7100 and D610 have this AF motor).
Besides the size of the sensor the number of pixels - and pixel density, AF features, speed, etc all make a difference. You should spend some time reading and watching (lot of Youtubes on this topic - some even good).
BTW - don't discount at D600. Find out if it has been served by Nikon yet or not - if not, Nikon will service it free (think of it as a recall). Some (not all) had oil spot issues which is corrected by Nikon. But there is very very little difference between the D600 and D610. Because of the oil spot recall the D600 is not favoured so will sell for less but could be a great bargain The D7100 on the other hand was a big step up from the D7000 (which still is a good camera but not as good as the D7100. I'd consider a D7100 or D7200 in the future for me, but not a D7000.
