Your Firmware Version: 1.0.0 Build 20160527 Rel. 60757 is already the latest one, the one you are trying to update is an old firmware. For the drop issue, it's better to contact TP-Link support to...

Did you try admin? If it doesn't work, hold the reset button at the bottom for 5 second, then try login again.

There is no place show uptime, you can find the time on the System Log. Why do you want to know that?

The wireless extender doesn't have "sleep mode", are your devices connected to it by cable or wifi? When you lose the connection, what about the lights on the two powerline units? You can try...

On the router's management page, go to Wireless--Wireless Settings, then you can see Channel, try different channels and see if it helps.

Before you do the speedtest with wired connection, restart your router first. Sometimes I got different speed before and after the router reboots.