#1 Battery
I have done this a few times, sometimes inadvertenly! I did it this evening, unplugged the AC power whilst still connected to the wireless network and it virtually drained in front of my eyes. Should I run it down when it's disconnected from the network? I reconnected it via mains supply and it's now 100% charged.
#2
Only over the last couple of weeks, it has been very slow to connect when on mains.When it picks up the network, it's fine. As I said on the original post, I tried another laptop, on battery power, and it hooked up immediately.