I have an iPhone. The same thing is happening to me, though this isn't anything new. My game has always lagged, but I've noticed that it's worse after clicking on the checkmark above certain characters to complete activities (Buzz and Sully being two of the worst). I almost always miss the first part of the dialogue when a quest activity is completed because I will click on 3-4 characters and THEN it finally pops up. Just in time for me to try clicking on the next one.
Battery has also been an issue. I thought it was just my phone, but it drops about as quickly as yours when I play this game. One time, my phone was on 48%. I opened the game and was on for barely 2 minutes when my phone died.