Mac OS X 10.10 doesnt recognize 1 and 3-finger gestures
One Queston, then my suggestion. 


  When I connect a standard PC mouse, I get the same outcome - 2-finger
  tap, zooming, swiping work. But not clicking.


Did the machine respond normally to the mouse click?



Given your description of the issue, here’s how I would proceed:


Make sure your software and firmware is up date. Apple Logo > About This Mac.. > Software Update
Disconnect all peripheral devices. Hold down Shift at the boot chime to enter Safe Mode. If you can’t recreate the problem, it’s software related. The most likely culprit would be a third party driver that's interfering with your trackpad functionality. 
Run Apple Diagnostics on the machine. Do so two times, back to back, regardless of the result. The machine must be plugged into your MagSafe. If the machine is not plugged in, you may receive a false positive that there is an issue with your battery. If you do receive an error code, follow the on screen instructions. Make an appointment to bring your machine in for a definitive diagnosis and servicing. 
Follow these instructions to reset the SMC. Components such as the keyboard lights, fans, battery, power supply, and trackpad all rely on this controller. If one is behaving abnormally, a reset may do the trick. See if the issue persists. If it still does, then proceed to reset the PRAM. 
Make an appointment with Apple to bring the machine in for a free and definitive diagnosis. Remember, your machine is covered under warranty. The last thing you want to do is void that warranty. Especially when you can’t even pinpoint the problem. Simply unscrewing the bottom enclosure will void it if detected. So, do not try a DIY fix. Bring it in. If it’s a hardware problem, they’ll fix it free of charge. If it’s a software issue, they’ll proceed accordingly (see below) until the problem’s solved. No matter what the issue is, as long as it’s under warranty, they assume responsibility for fixing it until it’s fixed. I don’t mean to come off as preachy, but I can't stress this point enough. 
I see no purpose in reinstalling OS X before diagnosing the issue. It’s the first step Apple will suggest to take after the hardware checks out. Not before.  However, if you do so, I’d recommend a clean install (wipe the machine completely, then selectively restore your data). 


Best of luck.