How to make objects precisely rotate and match the angle of random diagonal lines, and make it snap

Zernolf,

I would also do as Peter says.

Smart Guides can be your best friends, especially when they are allowed to.

Often, Snap to [Whatever] is overused, especially Snap to Grid as mentioned by Monika, which will wreck the use of Smart Guides.

And Smart Guides work fine without other snapping, just their own.

With the Anchor Point as shown by Peter, you can see in his screenshot that Smart Guides say anchor when you are within snapping distance; when you are within snapping distance of the line (and move along it) they say path. (until you get into snapping distance of Anchor Points or intersections or whatever).

You may also consider which Smart Guides options/settings suit you best (and omit unecessary/confusing ones).

If you need the other corner of your box to hit the other line exactly, as may be the case in your #3, you can do (almost) the same, only (kind of) reversed, as follows:

1) Select the red line and Object>Path>Offset Path with the offset equalling the box height,

2) ClickDrag the box by its lower right corner to snap to the intersection of the red offset path and the black line, Smart Guides say, guess what, intersect;

3) Set that corner as the centre of rotation and ClickDrag the opposite corner to snap to the red line, Smart Guides say path;

4) Delete the red offset path.