Photos

Ideally the best bet would be to use Google Photo for this and sync them to the cloud, then you can have access to them on your new device, computer, and any other Android or Chrome device that has your Google account setup. Plus the added benefit of if the worst case happens and the phone gets lost/stolen, broken, or the photos get erased somehow, you will always have access to them.
 
Yes -- in fact, I just did this yesterday with the new MXPE I bought my wife for Christmas to replace her 2013 Moto X. But be aware that the transferred photos will probably be placed by default in a separate "album" within your MXPE's Gallery. So don't be concerned if, when you open your Gallery on the MXPE after the transfer, the transferred photos don't show up in your main "Camera Roll" storage -- just go to the menu in Gallery, open "Albums," and see if they are located in a separate album (which might be labeled "Pictures"). Good luck!
 

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