I see you have thermal relief enabled on your pads, which means all the current will go through the four tiny 0.3mm bits of track around the pad in your input connector. It could be a good idea to use "direct connect style" on your polygon to avoid this.
Make sure the screw connectors can handle the current, it's in the specs.
If there are two connectors on the board, is this to daisy chain them? For high current it would be preferable to just route wires to the power supply from each board. You can also put the power connector in the middle instead of in a corner, current will have less copper to go through.
Don't worry about the copper, it'll work fine, resistance of 1oz copper plane is about 0.5 mOhm per square.
You can put several vias on the ground pin of the connectors: Place -> Fill, draw a copper rectangle, set its NET property, then place a few vias in it, for example on the side of the USB connector pin, you've got space for 3-4 vias.
Also you should rotate the fuse 90° and put it where the "R5" silkscreen is. Or you could put it at the side of the USB connector (with a wide trace). This is so the soldering iron can reach the pins of the USB connector without trouble.
These USB connectors have two thru hole pins to hold them to the board but they're pretty far away from the edge, which means quite important leverage, so they'll be fragile. You could use ones with the thru hole pins closer to the edge, or for DIY put 2 groun thru hole pads on either side of the connector and solder a bit of wire to strap the connector down on the board so it doesn't lift when it gets brutalized a bit.