Will my apps still auto-update if I root? I'd like to root my Nexus 4 with rasbeanjelly because I heard it increases battery life. If I do that, will the apps that I have installed still auto-update from the Google Play Store like they do now?
Yeah they'll auto update as long as you have enabled it under settings in Play Store. 

Keep it in your mind though, rooting or installing custom ROM will void warranty.