I have bought stuff from China and fortunately never had any problems with any of them. Its usually tested but at the end of the day its your luck. If you don't want to take any risk, buy something that is available locally but mind that it is going to be much more expensive.
I bet if you mange to find locally made programmer which supports as many devices as this one for $42! Its unbeatable price for a universal programmer. Trust me.
Yes you can do lot of things with this like read the chip, copy data to other chip, save device content to file, verify with other device etc.
No! PICs have internal FLASH ROM, they don't need external EEPROM for storing program. However serial EEPROM's are used with PICs to store data.
Old 8051's uses EEPROM. But new ones have internal FLASH so they don't EEPROM need any more.
And any microprocessor (which does not have internal code memory) needs an external EEPROM.
The programmer can program lot of 8051 variants so if in future you decide to learn that, you can use it.
It also supports serial EEPROMs (you will need in future projects when you want to store some user data) and PALs and GALs which are for advanced projects.
Again you are putting your hands in a very old technology so don't expect it to be future proof!