the shift-click is, IIRC, the loop/ring selection. so if it doesn't work, it means you're trying to select edges that aren't in the same loop/ring. You need quads to pull this off.
panel loops might be part of what you are after. It has polish and bevel features. It's pretty awesome tbh. https://pixologic.com/zclassroom/lesson/panel-loops-options-part-1
You ought to be able to insert a control loop just in the area of the extrusions and then use an ngon or triangle to terminate that loop. A few more segments will help but it isn't necessary to use an extreme number.
Looking much better with the loop in there, I personally feel the lifting of the minigun after the loop seems a touch linear it moves pretty much straight up. Just my personal thought on it :)
My first recommendation is removing any lines that loop around and doesn't contribute to any of the box's shape. I see about three loops you can delete right away.
Yep add some up n down loops from the corners of the middle indent. For hipoly models, correct shading is way more important than a couple extra edge loops.
It's that simple. If it isn't working, I don't know what to tell you. Vert spacing along the loop doesn't matter; as long as it's a proper edge loop it should turn it into a planar circle: