Screen size has nothing to do with resolution. Any size screen can run higher or lower resolutions, up to whatever the supported maximum is. If the older unit had a 720p resolution and the newer one has a 1080p resolution, then yes, you might take a hit on FPS. That doesn't account for any of the other issues though.
The Y50 is known to have overheating issues during intense extended gaming sessions, so it could be related to that, but doesn't necessarily have to be.
I'd start with doing a clean install of the GPU drivers, by which I DON'T simply mean installing the latest ones over older drivers or just uninstalling and re-installing the drivers. Following the steps in the CLEAN driver installation section at the following link, which does also apply to notebook GPU drivers as well as desktop versions:
http
/www.tomshardware.com/faq/id-2492424/laymans-simple-guide-solving-graphics-card-issues.html
Make sure you have the latest BIOS and chipset drivers installed per the Y50 product page. Make sure you reference your FULL model number as there are many sub-models of the Y50.
Run an SFC/Scannow to reduce the possibility of it being an OS issue:
http
/www.eightforums.com/tutorials/3047-sfc-scannow-command-run-windows-8-a.html
Run Seatools for Windows, run the short drive self test and long generic, to determine if it's related to the storage drive:
http

/www.seagate.com/support/downloads/item/seatools-win-master/
If none of that resolves the issue it may be a more serious hardware related issue such as a bad motherboard or faulty RAM module. You can run Memtest to check the RAM, but it doesn't always mean the RAM is ok just because it passes. In almost every case however, if it fails to pass memtest, there is an issue with one or more modules. If your laptop has more than one memory module installed, test them individually as testing multiple modules often results in false errors:
http

/www.memtest.org/
Last but not least, make sure you don't have an infection by following the directions outlined in this tutorial. Simply "having an antivirus installed" may and usually is not, enough:
http
/www.tomshardware.com/forum/8263-63-simple-free-guide-removing-malware
It may even be necessary to repair, refresh or reinstall windows, but I'd leave that as a last resort.