I have the same keyboard disappearing issue. If you REPLY WITH QUOTE the keyboard stays up. If you use the desktop version of IE it isn't a problem. I think this is a website issue. I emailed WPcentral about it but haven't head anything back. I have verified that the same thing happens on a Surface RT and a Dell Venue 8 Pro which runs full Win 8.1.
Another one that's been annoying me, not hugely but still. When you connect the 2520 using the hdmi the screen has to be on all the time, you would think you could select an option to turn of the display. Plus I wish the 2520 could do what the surface does when you connect it through hdmi, where it displays the desktop on the tv PLUS you can use the surface as normal so its doing two things at once.
I haven't tested the 2520's HDMI, so I may be wrong, but you should be able to go to the Devices charm -> Project and you can select your different display options there, ie. 2nd screen on, tablet screen off, etc. etc.