I am so sorry to hear of the issues you are having after the software update. Updates should improve your phone not cause issues. Let's dive a little deeper into this issue. Does the battery only get hot when doing certain things on the device? When do you notice that the phone is lagging? Please provide details
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I had the same lagging problem with my Asus Nexus 7 tablet after the Lollipop upgrade. I made the tablet nearly useless. After several weeks and many complaints, Google pushed what they called a fix for the lagging problem, but it didn't fix mine. As it turns out, my version of the N7 isn't compatible with Lollipop. Google should have known that before they pushed it to older N7's.