Engine turning on more often, runs rougher

I recently had a number of issues come up with my 2007 Prius, which I'll mention at the bottom of this post. But to be concise: the engine tends to come on now when I turn the car on, no matter what. When I first bought the car, that engine never seemed to kick on unless I got up to 30mph, or unless the A/C was on. Now, it's always kicking on.

It also will turn off at a stop sign, then turn back on again, sometimes even a third time. Also, I don't remember really feeling the engine turn on and off before, and now it's really noticeable.

So my question is how to diagnose and resolve this issue. Below are some more details about the car:

1. I parked it on a weird angle, some months back, and that night I had rough idling and a 'check engine' light. The code said a sensor was in the wrong position, but the code went away on its own.
   
   
2. I went too long for an oil change, but only a couple weeks. One morning, I got the big red exclamation mark, towed the car to a dealership, they said the battery was shot and the engine had a misfiring cylinder, and that I was down to a quarter of a quart of oil.
   
   
3. I towed the car home, changed the oil and filter, had Green Bean Battery put a refurbished hybrid battery in it, and the car has run without an engine light ever since.
   
   
4. I recently took the car to a mechanic and told them about the rough engine coming on all the time. I told him I was also feeling abrupt shifting, which is strange because the transmission doesn't have distinct gears. He said a solenoid was going bad on the transmission, replaced that, and the shifting issue is fixed, but the other problem remains.

 

These are great answers. I appreciate the help. Sorry, I did not receive any email notifications, or I would have responded sooner. To answer your questions, in order:

• There are about 150k miles on the car.
  
  
• I'm not sure if there's a change in MPGs. The display says the average is 41mpg, and I've been driving on the new, refurbished battery for 6 months.
  
  
• I don't have the Dr. Prius app, but I'll get it.
  
  
• I've owned the car for about 3.5 years. It had about 106k miles on it when I got it.
  
  
• I will check the SOC indicator the next time this happens.
  
  
• I don't remember the exact engine code but it was very close to this:
  P0024 OBD-II Trouble Code: Camshaft Position B - Timing Over-Advanced or System Performance (Bank 2)
  
  This was before I replaced the battery, though.
• The 12v is only 2 years old. I bought it new and replaced it myself.
• I will check the paperwork to see what the mechanic did with the solenoid.
• I will check the engine codes and post back.
• The rough running is roughest when the engine stops. Also when it starts, but especially when it stops.

I will knock out my to-do items on this list, and I'll check my notification settings to make sure I get any replies. Thanks so much for your help.