The battery issue with Nexus 6 is that after a period of time, the battery becomes damaged due to overheating. I took my Nexus 6 apart and the Lithium Ion shell had become deformed and the phone would cut out randomly in use and every-time when the battery got to 34%. I ordered a replacement battery for approx 15 USD and fitted it myself. I now have no battery issues and the phone is stable again i.e. hasn't crashed once.
I wouldn't advise replacing the glass yourself. It appears to be a complex process as the glass is glued to the LCD display. If the LCD isn't cracked, it is worth a try. The alternative is to replace the LCD digitizer entirely. Currently the LCD digitizer cost approx 180 USD and fairly easy to install, providing you follow the guides and have the right tools. I would advise you to replace the LCD digitizer, buy a glass screen protector (the replacement digitizers from China are not Corning Glass) and a new case to better protect your phone.