A:AnswerYes. The FAA doesn't want extra batteries in checked baggage because they can pose a fire risk. Portable battery such as this one are allowed in carry on always.
A:AnswerHow long was it plugged in. Depending on the charger sometimes it takes up to 12 hours to charge fully. Other chargers it just takes 3-4. I would say try a different one.
A:AnswerStraight out of the box without charging it first it lasted a few weeks. I charge my iPhone a couple time a day with. After a few weeks it went down to half finally so I decided to charge it.