I've only had this tablet for a month, so I am not the most experience with it yet but here we go.
There is a couple of threads already on this forum where you can discuss common issues that you can check out.
About the battery and charging - i have heard that asus reccomends not keeping the tablet/dock plugged in to power all the time, I usualy charge it during the night when I am asleep, for me it easily lasts all day even without any battery saving options on but that may vary on usage
When you have the dock plugged in, you can see its state and battery percentage in the notification tray. For percentage of the internal tablet battery - you can go to settings: under battery there is an option on the top to enable battery percentages (taking you are on the latest stock update 4.3)
You can find settings for the trackpad such as speed, pointer type and button config under "Settings > Language&Input" on the botton theres a Trackpad section
I can not give you a huge list of keyboard shortcuts, but here are the ones I have found and use:
Alt+Tab works as you would expect - a fast application switcher (i use this a whole lot)
Fn+Arrow Keys - Page Up/Down and Home/End (also very useful)
As or the menu key you asked about, it is the hamburger-looking key on the right-hand side of the spacebar between Alt and Ctrl (might depend on the local keyboard layout?)
The shortcut row on the top row of the keyboard I think is pretty self-explaining
I'm not the best explainer, but I hope this was somewhat helpful
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