On my note 2 with dn3 v4 beta 2, the downloaded apks from my playstore are in root/factory, different devices may have it in different name.
If you are talking about apks downloaded from internet or transferred from PC, then it most likely will be in root/sdcard/download folder.
I think your situation may be a little different than OP's. From your sig your gear is rooted with null rom 26 (btw where did you find that, I thought the latest null rom is only 23?), so you would have a custom recovery on your gear already. Then all you need is to load the apks through bluetooth file transfer on to the gear, and intall it from the gear directly, you don't need gear manager at all for this.
As for why your s3 is not recognizing the gear, I am not sure. I had my note 2 on dn3 kitkat with latest gear manager (1.7 something I think) connected to my gear on null 23 rom and everything was working, including bluetooth file transfer, and I was able to install apks by bluetooth file transfer and then install from the gear itself.