RE200 must use the same wifi password with router and it can use different SSID with router.

Don't hear that they have compatibility issue. You can log into the management page of 850RE to configure it, this is easy and there a a lot of guide.

I also think wired connection should be able to work.

Where have you posted?

It seems that this device is bad unit, it is recommended to replace another one first. When you connect PC to RE210 via ethernet cable, can your PC obtain IP address?

This has something to do with the operation mode of 850RE, anyway, it is normal. If the devices would disconnect for a while, maybe you try to to upgrade the newest firmware on TP-LINK UK company...