For now, you can caulk the holes in the siding, but eventually, think about replacing it. It will continue to rust, and create bigger problems down the road.
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For the door I would make it overlap so when you close it there is wood behind the door so when you close it the crack is covered on the backside, you could do this to the top and bottom, dont know if it would work well with that door though and it would probably be best to make a new one.Then make sure it closes tightly against the wood and then latch it. For the holes in between the rafters use hardware cloth or cut plywood out to screw in.
The door may need to re-hung that door to make it sound. Depending on the framework of the door, if its sound, a tad larger screw, or fill the old holes with a dowel rod, cut it flush with the jam, and put the screws back in. If the frame isnt sound, and has some rot, you will need some new lumber for that to re-hang it. That should take care of that gap. How far up are the soffits? If they are above the roosts, just cover the gaps with hardware cloth. But, if you have some that will cause a draft lower, fill those with 2x4s, and nail into place. Depending on the overhang of your roof, you may be just fine with the hardware cloth.