You can get t-posts for cheap at tractor supply, and use cattle fencing material for the walls, and if you get the ridgid cattle panels you can easily slide a few 2x4s across the panels and make a weather resistant area. You dont say exactly what you want, or the weather the animals will face so it is hard to go further. https://www.instructables.com/Cheap-Easy-Portable-Dog-Run-to-Build/ has a ton of different things all diy.

You might want to double check your agreement whether it allows you to build something like a kennel.

I want them to be comfortable in both the very cold (the lowest it was this winter was in the low single digits- even the water in their bowl froze inside- only heater I have is crap), and the very hot- I think I already mentioned triple digits are.common.once summer really starts up?

Had to look up what you meant by cattle fencing- didn't know that was the name for thestuff. About how many feet would you recommend I buy? Depending on the cost, I may just get one roll to start with, expanding when I can afford to

Ah, very true! With the not DIY kennel being allowed, I hope that means my DIY one will be too- but it's better to ask first just in case

In winter it rarely snows enough for there to be more than a very small layer of it covering the ground- but in the past...5 years maybe it snowed enough on two separate occasions that the areas I lived inat the time pretty much shut down.

We do sometimes have bad.storms come through, wind afaik hasn't reached hurricane level speeds as far inland as I am.

If I want to go the buying used items route for some/all of what I'd need, it seems to me that living in an area where there's a fair number of people that raise cattle may have some older but still useable things I could buy from them

Asking my.mom if it's ok for me to set up something on her and her fiances property- it's owned by them, so no pesky landlord to need permission from

So central Texas style weather? So sun shade, wind shade, rain protection... Cold but not for long - you could do the diy route with cattle panels and then make a plywood and 2x4 box with a slight slope to the top to shed water. Solar screen material from home depot/lowes can shade it really easy and is cheap, They also have a waxed canvas tarp that is water proof and ready to hang on the kennel.

How would I go about making a gate so I can go in/come out?

Then you will definitely need some insulation too.

Any suggestions on what to use?

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Ask for insulation material at the store, it's a special material. You can also use pieces of foam, or spray foam. You will of course need to sandwich whatever material you get between solid layers; one layer to the outside but also one on the inside (foam can disintegrate if not protected, and insulation material can cause rashes).

Keep in mind that insulation should go in all the walls as well as the floor and ceiling.

It's also important to raise the dog house off the ground so it's not in direct contact or it will leech heat like crazy. You can put it on anything, bricks, pieces of wood, just make sure it can't fall off when the dogs bounce in and out.

Moisture isolation is also important on the floor side, a layer of tarp would be good. Also some kind of mat or rug wouldn't go amiss on the floor.