"Can the rotted wood on the door where it’s supposed to meet the hinge be repaired or do I need a new door?"

It can be repaired inexpensively. Google Abatron please.

Even cheaper--replace that bottom piece of trim. I would remove the shutters and add trim around the windows while I was at it.

It is a small shed. No need for a power washer. Scrub brush will work fine.

I can't tell from the pic how much of the door and/or trim is rotted and no longer structurally sound so I lean towards replacing the trim and maybe reinforcing the door if rotted as well. Abatron and other wood repair products are great (but not the cheapest or easiest route). If only the trim is rotted but the door is sound, definitely go with replacing the trim.

I agree with above, replace that wood trim, thats rotted, and then clean the building with a scrub brush, to get it nice and clean. I would keep the shutters. Use a bit of bleach in the water that you scrub with. They make the shed look much nicer. Repaint the shed and all the trim and shutters, if its necessary after its cleaned.

If water damage is likely to recur, consider using pvc products where you can instead of wood. They are not structural, but they are waterproof.

I recently spent $4K having someone repair wet wood on my house. We used a combination of products including PVC ”lumber”, wood epoxy, pressure treated wood, acrylic stain on cedar siding, other waterproofing materials he had, a new vinyl window for the hard-to-access upstairs bathroom window, and lots of caulk. We also restructured a few things to prevent future problems. The house is pretty much water resistant now.

Think about things like gutters and drainage and sloping ground borders to prevent future water damage. If that’s impractical, maybe some kind of waterproof bottom border on the building might help with splash-back.

You can also look behind the shutters to see if there is water or insect damage there. If they are plastic and there is no damage, you might keep them. If water damage, remove them. If insect damage, consider replacing with solid shutters and closing up large crevices so insects cannot get behind them. If they are louvered wood, they might deteriorate. You can replace them when they start to fall apart, or when you don’t feel like painting them.

I think solid shutters would look better. But make sure no water is likely to get behind them. Or maybe just some wide trim around the windows might look better than shutters.

Regarding cleaning, you could take a stiff brush and see what you can scrub off. My house has cedar siding and I did not want to pressure wash it because the wood is soft and old. I scrub it with a stiff brush and remove the residue with a soft brush, then apply solid acrylic stain (Cabot) over a dry surface. It worked really well. My house was never painted. If you are considering painting, stop into a Sherwin Williams store and ask for advice on cleaning and repainting. They have lots of products for different needs.

Looks to me like everything could be repaired. If you have moss on the roof, that's a problem, too.

Sadly nothing has been invented that allows repairing sheds with strokes on a keyboard or sprinkling with magic dust. Remove rotted wood and replace with sound wood then paint. Water hose,bucket of soapy water and a scrub brush. Leave the shutters and paint the shed. That's all there is to it and your shed will look great.

The reason that I would remove the shutters and install trim around the windows is because the windows are surfaced mounted with the fasteners through the flange making it look unfinished. And because the siding under the left side window is pieced and some of the sill flashing is exposed. Simple flat trim will hide a lot of the ugly. Plus the shutters are crowding the door.