Any battery has a limited number of cycles. It will be doing micro discharges/charges that eventually add up to multiple cycles.
Heat also plays a role in diminishing battery life.
Probably a long term thing, but heat can also play a role in degrading the rubber of the keyboard.
Simply put, I wouldn’t. But if you play it frequently enough throughout the day, that the 30 seconds it takes to boot up is an inconvenience, arguably, the initial boot up of the device would cause more strain than having it constantly on. I’m talking like, turning it on and off 30 times a day for a year.
Re: batteries, I can’t remember what kind of battery the game shell uses, but as a general rule, for any old phones, laptops or anything with a rechargeable lithium ion/polymer battery, if not using it for an extended time, store it with 50% charge.
For the record, I have a rPi unit and a small 7” display connected 24/7 just running arcades in the background. It’s a throwback to my life as a kid, waiting in hospitals a lot, staring at arcade machines doing their demo loop, wishing I could play them.