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You can drag to select the line, but actually “grabbing” it with the black arrow tool in order to drag-move it to a new location is nearly impossible - you just wind up missing its visible area and deselecting it or starting a new rectangular area selection. This is especially true when its center (handle), or any other anchor point, is off screen. When an anchor point is visible, you at least stand a chance to grab by it(s little colored square area which occupies some space on the screen) and initiate the drag operation.

It is very difficult to drag move a thin line without zooming in, in order to give it some “pixel width area” to have something to grab by with the black arrow.

Having its bounding box visible or not does not seem to help with this endeavor in any way.

I am often forced to zoom way in to give the thin line some screen width to grab by or using the Layers window to select it from there, both of which are a serious pain in a complex illustration.

This is even more frustrating when dealing with 0.25pt lines on a 4k 32-inch monitor at reasonable zoom levels.

There must be an easier way to drag move something that is very thin (and selected, if necessary)!

"There must be an easier way to drag move something that is very thin" --- I could not possibly agree more. I waste so much time with click-drag, click, miss, click-drag, click, miss, click-drag, click, miss, click-drag, click... move.

However, unfortunately, Illustrator has suffered from this problem since at least version 8 (not CC8.. actual version 8 from the 80s/90s). Using the Preview Bounds in the Preferences can sometimes help. But for other reasons I find working with Preview Bounds on to be more trouble than it's worth.

I don't think there's a workaround other than using a lower screen resolution, which is a horrible alternative. People have been complaining to Adobe about this for over a decade.

The only possible alternative I've found is to use some of the AstuteGraphics.com plug ins ($$) which have selection options if they will work. For some reason AstuteGraphics has been able to make selection much easier than Adobe has. Or, use things like the transform panel to move objects or arrow key nudging -- which are all less than optimal solutions.

You may be able to use View > Outline (Command/Control-Y) to go into outline mode and find a section of the line and select. I often toggle it on and off to make fine selections.

Depending on what Tool you have selected at the time, that's what is supposed to happen. The black arrow selects an object as a whole, the white arrow selects just one point (or line between two points) at a time.

If you just want to be able to scroll your mouse around the active Illustrator project without it effecting the lines/curves you've created, lock off the objects in question in the Layers window. This will prevent any interaction with the object, of course.