Just checked that link and it seems I already have the most recent driver.
I've just noticed on my screenshot that the wi-fi was disconnected when I took the screenshot - this must have been one of the disconnects as I was using both ethernet and wi-fi at the time. I've attached another screenshot with the ethernet unplugged.
There's nothing wrong with your Wi-Fi connection either.
I shouldn't worry about not being able to get simultaneous Wi-Fi and Ethernet connections.
Your network is identified as being Hybrid in both of your screen shots but you would need a hybrid router in order to achieve a Hybrid connection and the home hub is not hybrid. Therefore you will only get one connection at any one time and the default is a wired connection (as long as an ethernet cable is connected).
Can you get a good, steady and reliable connection on either ethernet or Wi-Fi when the other is disabled or unplugged?
Unfortunately no I can't get a stable connection using just ethernet or using just wi-fi. The problem happens when using both, one, or the other.
I can't see anything wrong in either of your screenshots which makes identifying the problem more than a little difficult.
All I can suggest is that you keep one eye on the networking icon in the system tray in the bottom right hand corner of your screen and see if this changes or you get any on-screen messages when you lose connectivity.
EDIT: Have you tried re-starting your machine
Does the problem happen when you are using the laptop plugged into power and when on battery power or does it only happen when on battery power?