I have the very same problem, but in my case it was caused by a wrong charger.
My mother, as an outside-technology person she is, wanted to try some random chargers she found around so she plugged them into my S2. The phone shut immediately on the 2nd charger and has been malfunctioning ever since.
-Battery lasts for 15 hours on standby.
-It says it's fully charged within 20 minutes on charger, no matter what.
-It keeps showing the "Battery full, remove charger" icon even when not plugged.
-If it's not plugged on the charger, it's not booting normally.
>sometimes just shows the battery icon with a yellow warning triangle-unable to boot.
>sometimes it boots but doesn't recognize my SIM.
>Phone menu (Power-OK-Vol Up) menu always works so I can still reboot it.
-It has recently started showing a USB icon since it starts up.
So, I have to keep my phone on charging all time I'm home even though I know it's bad for it. I've got no hope for any USB usage so I use bluetooth/Card reader in order to transfer files. My guarantee period ended 40 days ago and I was stupid enough not to send it for service before it does.