Welcome to the forum, glad you joined.
My first consideration in planning a coop/run system is to look at drainage. If you build it where water drains to it, or even worse stands, you are almost certainly going to have issues. If water drains way from it your life is easier. If you want to discuss that more let me know. I'll be watching this thread.
My personal preference is also for a walk-in coop. My grow-out coop is a 4' x 8' elevated coop but my main coop is a 8' x 12' coop on the ground. It's just so much easier with that walk-in coop. But if you limit yourself to 6 chickens you can make an elevated coop work. And don't think of the coop in isolation. Think of your coop and run as an integrated system. Your climate and how you manage them has a part to play in when that space is available. You said you get a lot of snow. That argues for a larger than normal coop or you enclose your run enough top and sides to keep snow out. That might make the run expense.
You have a small budget. I totally understand that. If you buy your materials new most building materials come in 4' and 8' dimensions here in the USA for less expensive sizes. Not sure where you are but I suspect it's similar. If you take that into consideration you can avoid a lot of cutting and waste. A 6' width isn't horrible. The cut-offs can be used for nests (for 6 hens you will want 2 nests) and the roof needs to be sloped so water and snow doesn't stand. Sloped means they are longer than the width plus overhang gives you a good place to put ventilation that keeps rain and snow out. If you have an alternate source of materials use the dimensions of that material in your plans.
If you are in the USA you probably have access to Craigslist, if somewhere else you may have something similar. With a little patience and some work you can often find an existing out-building that you can get for free. You will probably have to transport it yourself or maybe even tear it down and rebuild it. Or use the materials for another design. Sometimes you can get materials really inexpensively or just for hauling them away. If you are looking for a design you are probably handy enough and have the tools to turn a shed into a coop. Anyway Craigslist or something similar is often a good hunting ground for inexpensive stuff.
Hardware can get expensive. In addition to Craigslist, do you have a "Habitat" store around. Sometimes you can get some pretty good deals on hardware there. Sometimes windows and doors. If you have one it is worth walking through.
Don't be shy about asking for scrap materials at construction sites. Often they have construction aide materials like forms for concrete or lumber to hold things in place temporarily that they are just going to have to pay to haul to the dump. Other than some possible embarrassment if you are shy, what will it cost you to ask?
Not much help on actual design, I know. I'm trying to think inexpensive. If you follow the links in Aart's signature in the post above you can get some hep there. I wrote that "Space" article. My general recommendation is to give them more than the minimum space. I find the more I crowd them the more behavioral problems I have to deal with, the harder I have to work, and the less flexibility I have to deal with issues. I do value flexibility.
Good luck and again,
