HSDPA/3G not working and Power off when plugged to charger
This could possibly be software or hardware related. If you power the phone down first and then plug it in, does it start charging or does it do nothing? If it doesnt, I would start to lean towards some type of hardware failure (hard to say for sure though). Have you also tried a factory reset on the phone? If it is software related for some reason, then a factory reset may fix that up.

Did this issue appear out of no where, or did you recently install new apps or something when the issue started?