At lunch, so will try to come back to later if there's anything else...
Tablets generally don't do Flash, whether Apple or Android devices. Lots of game apps out there, though, just be warned most are freemium (so avoid storing payment details on the device, and beware windows in which other payments are authorised after paying for one thing, etcetera).
For your suggested needs I'd probably not look further than Amazon. There's a 7" kids edition of the Fire --
http://www.amazon.co.uk/Fire-Kids-Displ ... 00YYZMF1Y/
If you want big, the Fire HD 10 and other models are also decent value. Amazon load their own variant of Android whereas I was after stock Android, rooting, etc. and have gone with Nexus devices.
Amazon are obviously hoping people will continue to buy into their subscription services after a year so the devices are decent enough in their own right and essentially partly subsided. And you can use microSD cards with them. I suspect that's going to be your best starting point.