Kumar, ditto on "edit and remove email address" in your first post. This is where you can get help and ideas for designing your back yard, but not really where some one will hand you a plan. There's a fair amount of work involved. You'll need to provide more information in order to get help and be willing to assemble the information you get back into some sort of plan. One picture does not really show the extent of the back yard. If you stood next to the center of the back of the house and took overlapping shots while panning the yard (all shots taken from same position) then we could get a better feel for what is there now. In order to get control of weeds, you'd need all areas to be mowed turf, heavily mulched, or thickly planted. The combination that will work best will depend on your needs and desires, and the existing conditions. In order to get help in figuring out what arrangement will work best, provide the additional photos and more information about how you will use your yard.

Yes, and some of them look like evergreen trees, notorious for hogging all the sun and water. Pots of shade loving plants do work. I do mulch under my trees and pots of bright colored impatiens mixed with vinca and begonias. Lobelia is good too. As for your deck, I'm thinking you just got the house? I wouldn't rip out the patio if you need more space, you can add on to it. Start using it and figure out your needs. Things to think about: privacy and views, are there views you'd like to block or privacy you'd like to establish. That can be solved with landscaping. Shade can be provided by trees, there are some fast growing ones that are nice for patio areas like autumn blaze maple or shademaster honey locust, for example. Figure out the path of the sun to figure out where the shade will be cast. Are you going to grill and eat on the patio? You can start with patio umbrellas covering those areas and decide if you want a pergola. I would love one, but umbrellas will work and the nice thing about them is you can move them around as your shade needs change. I'd wait a bit before committing to a pergola to figure out how big and where to place it. How about views? What do you want to look out on from your patio? Do you want a play area for lawn games or the like? A fire pit area. A kitchen or vegetable garden. An area for birdwatching? Three season interest. A place to snooze in a hammock? Just some ideas. Start with some thoughts on how you will use your yard, which will develop as you live there for a year. Decide on a plan for the whole yard and work on it in stages. Start with the things that will make the most difference, like hardscaping and trees.

You can’t fight physics. If the dogs use this exact path every time they go out, you probably aren’t going to be able to get anything to grow there. I’d lay flagstones. Yes, you may just move the dirt spot further away, but I don’t see any other solution. Just get the spot out from the house and hopefully the dogs will “scatter” a bit and not follow a direct line to their tree.

Consider creating terraces. Decide how much room immediately behind your home you eventually want for a backyard patio. Leave at least six feet beside each fence not dug but dig a ditch/footing for a wall of concrete block down to hard ground across the center of the rest of the back yard. Once your terrace wall is built, level the area above the wall, including by adding a truck load of gravel and/or dirt and/or shoveling dirt from below the wall to above it to level both your patio area and a space below the wall. Once you've reserved your patio space -- while your dogs pack it from using it -- you can begin to plant your garden, preferably in a way the dogs can be taught to leave what you plant alone. Pay attention to how the sun passes over your back yard so you're not planting anything that will eventually shade your garden too much for it to grow. Brackets on fence posts can hold potted plants -- choose the strong ones. https://www.google.com/search?q=very+strong+bracket+for++holding+flower+basket&source=lnms&tbm=isch&sa=X&ved=2ahUKEwjy1KW5l4LoAhWiSt8KHRZkBw0Q_AUoAnoECAwQBA&biw=1366&bih=628 Consider the stronger tomato cages for tomatoes and beans. You can buy individual trellis/cages in the spring from the big box stores or build your own -- there are You Tube videos on how to build different kinds. This is just one. https://www.youtube.com/watch?v=ozQNsd5Oek4  Consider building a mini greenhouse using plumbing pipes and/or wood. https://www.youtube.com/watch?v=KsSI4w3CqEo https://www.youtube.com/watch?v=sQWPnaiixe0 https://www.bing.com/search?q=building+greenhouse+with+pvc&form=EDGSPH&mkt=en-us&httpsmsn=1&msnews=1&plvar=0&refig=d03dc932a900445e84174ddd49ea6bb8&sp=1&ghc=1&qs=AS&pq=building+greenhouse+with+&sk=PRES1&sc=8-25&cvid=d03dc932a900445e84174ddd49ea6bb8&cc=US&setlang=en-US If you build raised beds with wood, remember to line the wood box with plastic to make the wood last longer. https://www.youtube.com/watch?v=ef_TS19TRQg and / or building tiered structures for plants https://www.google.com/search?q=wood+tiered+structures+for+planting+strawberries&tbm=isch&ved=2ahUKEwiv8fyvk4LoAhXEEFMKHS1JBKMQ2-cCegQIABAA&oq=wood+tiered+structures+for+planting+strawberries&gs_l=img.3...95448.101579..102697...0.0..0.134.1724.15j4......0....1..gws-wiz-img.NbM-ukJ-fRs&ei=qFFgXq_9FcShzAKtkpGYCg&bih=654&biw=1366&client=firefox-b-1-d#imgrc=irj20Wqm9lLhnM