How do I repair/reattach downspout and gutter after snow/ice damage?
Well, fixing the damage won't prevent it from happening again, so to fix this and prevent further damages from ice in the future, you will need to install some hardware. First, you are going to need to get out the ladder and , with some assistance, push the down spout back up and re-attach it to the elbow. I'd use extra fasteners, and possibly a short rubber connection hose to ensure a good tight seal. It is likely ice formation caused the metal to flex, creating a weakness in the fastener, so the rubber hose will help.

The gutter issue really comes down to weight. Installing some "Heavy Load Gutter Brackets" would help. For the distance of gutter and weight on it I'd put in two brackets with each in the middle of the roof jousts near the middle that are shown in the picture.

With all of these fixes, be sure your nails or screws get "purchase" or else it is all for nothing.

EDIT:Use the same fasteners you are using, just use more. Not a ton more, maybe just one or two. Don't use sheet metal screws, use wood screws. Sheet metal screws have smaller thread, wood screws are made for, well, wood. That will handle the horizontal movement. For the vertical sliding, either rivet the joints together(this option costs less, but more permanent, and is more invisible) OR buy a  vinyl connector and put a fastener on the connector itself and on the downspout itself just above and just below the connector