It sounds like you have done enough to rule out problems with the cord or the outlet, which is a good start.
Question about when you tried the iPad: did you use the same cord AND wall adapter, or did you plug the cord into the iPad wall adapter?
The reason this is important is because in trouble shooting charging issues there are 5 physical things that need checked out: the outlet, the wall adapter, the cord, the charging port, and the battery.
Sounds like the outlet and the cord are good. Next verify that there isn't anything wrong with the wall adapter. If the wall adapter is fine, check to see if there is any debris in the charging port. You can use compressed air to clean it out if you have it available. Also try to visibly inspect the charging port for any physical deformity that might cause an issue, although if you can still plug the cord in just fine that isn't likely.
If you get to the end and there isn't any change and you can't find any specific part malfunctioning, then the issue is most likely something internal to the phone and will need repair by someone who knows their way around the inside of the phone (i.e. Apple or a legit repair facility).
As for turning off the vibrate when you plug it in, I've never heard of a way to do that, sorry.
