tablet won't permanently keep or remove apps
It sounds like you have a tablet with locked system/data. Kind of like a PC with deep freeze, you can make changes but they are reset after a reboot.

You may have just made an unlucky purchase as many cheapo tablets will do this as a sort of self-protection.

What is likely happening is the system reverts to a copy of the image stored in flash so fixing it is going to require finding wherever the source copy is.

The md5 file error likely means the original copy is corrupt. This is the tough way to learn to avoid cheapo tablets because when something goes wrong you are usually SoL.

Depending on the tablet, you may be able to:


unlock the bootloader
install custom recovery
find a compatible rom and flash it


I'd recommend you just cut your losses, sell it as defective on Craigslist or something similar, and use that money to put towards a decent tablet.