New graphics card causing internet lag spikes, but only while gaming

you should choose High Performance NVIDIA processor, this is suitable with graphics device of your laptop VGA card. About integrated graphics, this software installed automatically by OS ( WIndows ) and not match driver for VGA graphics device. Don't forget to check on Device manager to ensure that graphics device driver installed correctly.

So, considering the discussion in the comments, and my opinion, possible causes of the FPS drop and screen tearing for the specific game in question, CS:GO, would be (in order of likeliness)

• Corrupt configuration files on CS:GO. (OP already reset them, discarded)
• CSGO or Source Engine update issues (No updates were released at the time of the incidents)
• Busted drivers. (OP already reinstalled them so, discarded)
• Broken GPU fan, so it throttles down to prevent damage. (Op checked them, so discarded)
• Dirty GPU contats. (OP already cleaned the card, so discarded)
• Tweaking fps_max value. (Already done, discarded).
• Nvidia GeForce Experience causing issues. (OP uninstalled it, so discarded)
• Steam Overlay causing issues. (Op disabled it, so discarded)
• Windows update messing with OS. (Op stated that no updates occurred before the incidents)
• Nvidia HD Audio causing issues. (OP uninstalled it, so discarded)
• Busted Video Card (By OP's comment this turned out to be the culprit)
• Busted RAM (Unlikely)

At the time of this answer we don't have specific information about performance in other games, OP specified that the screen tearing was occurring on Borderlands: The Pre-sequel as well, but we don't have specific performance data. On the other hand Furmark stress tests didn't crash the card, but the issue in the games persist.

Given that information the only option we have left is a somehow damaged Video Card, the solution would be to fill a RMA, as the card was recently bought. I suggest contacting with the manufacturer instead of Amazon or Nvidia. The manufacturer (if professional enough) would give you a tested card, guaranteed to work, while Amazon will give you another boxed one or a refund if you are a regular client. Nvidia will do nothing they only build the chipset. I've had a reasonably good experience with EVGA for example, not sure about other manufacturers.