Plug the charger in at least overnight. Then turn iton with the charger still plugged in and see if it stays on for more than 15 seconds. (A Note 3 should run with no battery and the wall charger connected - that's how mine was set up in the store. I refused to take a battery that they had abused by using it before fully charging it. If the battery is
really dead, it may take hours to put enough charge into it to let the phone run from the charger with the battery still in it.)
If the problem is a bootloop, not a bad battery, you could try reflashing the ROM (there are stock ROMs on XDA that won't overwrite your data - look in
T-Mobile Galaxy Note 3 Android Development - XDA Forums for one, or ask there - a lot of the developers there think nothing of ripping a ROM out of a stock phone and uploading it for you).
If the pictures are on the SD card, they'll stay there when you put the card into the new phone.