Can't move a folder

Grouping to move is a good idea. Let's try some things. With your move tool selected, in the options bar at the top of the screen there is something that says Auto Select. You don't have it checked in your first screenshot. Check it,  and choose group - this will move all layers in a group together. You can also link layers. Let's say you have text over a button or image, if you shift click those layers to select them, at the bottom of the layers panel you can hide the link button and they will move and transform together. I don't see anything in there that should be giving you problems. You can also save all of those layers together as a smart object - then they will be on one layer in your document but you can go back and edit if necessary. There is a lot to learn. Is this your actual work? You should ask your employer to pay for a class. There are lots of places that do online classes for professionals and you are a perfect candidate and typical of my students :).  I do private training and contract with Certstaffix, but there are lots of companies out there - look for a 3 day intro to PS class. In 3 days you can go from frustrated beginner to frustrated intermediate - I mean intermediate user, lol.