You'd reduce the mud if you had your downspout drain into a rain garden or a pipe that is a lot further from the house. Gravel will work well unless you are going to shovel snow from it, but you should do a path with your existing pavers in a gravel bed. There are plenty of shrubs or perennials that should do reasonably well in a damp shady place.

Much of the space along the right hand wall in the top pic is too close to the house to walk on, so you'd do fine putting a few shrubs with a path around them. Or a bed of plants.

I agree. First things first: you'll have to get the water away from the area, or it'll never be better.

Fix the drainage. The easiest quickest way is to attach a drainpipe to the downspout so the water gets away from the house. Add dirt next to the house if you have to, to get the slope going away.

I think the mud is especially bad just right now with the rains and thaw. And the fact of no vegetation against the house or garage. I love the idea of shrubs against the house (to the right of the door in the photo) after we level the slope there. I think the blank wall to the left in the photo needs something as well.

Flagstone and gravel or crushed granite sounds really nice. Flo - when you say 4/6" of crushed granite, is that the depth? Does the flagstone nestle into the granite? I need to do some research on laying flagstone.

If you have loose stones or gravel, with patio stones embedded in them for walking, I think you will have problems with the dogs. They will drag the loose materials around and if they use them as a toilet, hard to clean.

I think you need a solid surface, either a deck or a patio.

Your husband is correct that a wood deck is slippery and hard to use in winter. Facing north, it can also get slippery with algae when wet, even in summer, although that can be scrubbed off.

A better deck surface is a synthetic wood. We have it on a wheelchair ramp, and it is great. Never slippery, doesn't ever need staining or scrubbing, and looks the same as when we put it in almost 20 years ago. More expensive than wood, but might be comparable to regrading and putting in proper patio with regrading, base layer etc.

Beautiful idea, but if you don't think you'd want to sit there because of privacy issues, you might prefer a something simpler.

deck slippery? why not use a composite decking material? (After addressing your drainage issues that Flo mentioned ). Since it's a north facing, during the winter that ground will always be muddy. you could do shade plants, but w/the dogs, why bother? even gravel will get messed up.

large concrete squares w/pebbles in between could work. But the deck would look the best.

I have one around my pond that sits just above the ground. it's not slippery, doesn't require maintenance, but can get a little warm in the summer. you could always do a patio cover above it if you need too.

not quite done with everything else (the walkway needs to be done, and the house painted) but you can see how the decking material is laid out w/the patio cover above it.

we lucked out on this decking because it was a left-over lot. it was close to a 50% reduction. check local places that sell this material and see if they have any on clearance (if budget is a concern)

Yes, you lay a deep bed 4-6" of the crushed granite with weed prevention under it, and then place large flag stones into the granite. You just need to level them into the crushed granite and let it naturally settle down. You then will add more crushed granite once things settle a bit. Just make sure the grading from the garage/house goes away from the building a good bit of a drop. I wouldn't plant right up to the house. Just creates problems with roots and debris. Put any plant materials on the outside edge of whatever shape you select. You do have to put a "edge" to define the crushed granite space so the granite doesn't migrate into the grass, etc. Most yard people know how to do this and it is quite budget friendly.

depending on what you want to spend or what you can DIY, ideas...

the concrete squares I mentioned. you can either buy them, or build forms to your size and pour it yourself

You all are so amazing with your ideas! I'm so grateful for your help. It's helping a lot to hear your opinions and experiences.

If the area is well drained and the surface is permeable - gravel, pavers with gaps not grout - then it won't be slick. If it's not well drained almost anything will be slick. Fix the drainage as suggested by other commenters and then figure out what makes sense for you.

For gravel, chicken and turkey grit is cheap and effective!

Regrade and deal with the downspout issue. Frame the area with metal edging and fill with 4 inches of pea gravel. Pea gravel allows drainage, won't ice over, because of its size very little will be tracked into the house by shoes or dog paws. An occasional raking to level is the only upkeep needed.

I think the majority of your money needs to go into diverting this water away from you house. I worry about your foundation with the land sloped toward the house. Regrade, divert water away, and take care of the downspout. Our pours directly into our sewer connection. Where do people in your area divert their down spouts to?

I wouldn't try to get too creative here. Put in a nice brick walkway. Get a few nice tall planters with an ever green shrub in each to have a focal point and greenery raised off the ground enough so the dogs won't pee on them.