Plastic bucket under gutter downspout

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

Personally, I'd cut the downspout back so it is even (level with the ground), with about an inch of pipe hanging down (or as much as can be saved). Get another piece of pipe, in which the inside diameter is the same as the outside diameter of the original downspout. This will be used to replace the piece you just cut off. Before you affix your new pipe to the old, you'll need to put something on what is left of old pipe to save it from deteriorating further. I'd use a rust converter, something like POR-15. Use it liberally on the inside and the outside of the pipe ... wherever you see deterioration. If the new pipe is a snug fit on the old pipe, you can actually use the POR-15 to bond the two pieces of metal together. You'll also want to put some type of screen up top to prevent the debris from getting stuck down in the pipe again, or you'll run into the same issue. You'll need to get some type of reducer which will allow you to transition from the new piece of pipe to the crimped piping which is there now.

This would be the "easy" fix, if you will. If the deterioration is too bad, you'll need to consider a complete replacement of the gutter system. This would cost some serious money, but would bet you are going to have to do it sooner or later anyway. Check any other downspout to ensure the same thing isn't happening there. Use the POR-15 to prolong your pocketbook agony for as long as possible.