Okay, I'm going to try to help you here. First, try to decide what you want your chicken facilities to look like "one day." I hate to have to re-build anything. Next, figure out which part you can build and use, first. I think if you build a nice, secure run, it will serve as your coop until you can get a building put up. Like you said, you can tarp part of it and use it that way temporarily.
You said your two areas are 5x4 and 5x10 so I assume the footprint will be 5x14, correct? I propose you make it 8x16. Now, before you say no, we can't afford that, think about it. Standard lumber is 8' long. If you make it 5' deep, you have to cut lumber to 5 feet and you have a 3' scrap. Make it 8' and there is less waste and the 16' width is 2 standard boards.
Now, lets talk about the supports. These are the bones of your structure. I see you planned on PT 2x4s but you will have to sandwich them on the corners to make sure they don't warp or you will have to use more of them closer together to make it strong enough to support a roof. Four PT 4x4s on the corners (or, ideally, 6 so you could have one in the middle of each long wall) would be stronger and likely cheaper than 8 2x4s. Check your area for a sawmill or lumber yard that might be less expensive than your local big box hardware store. Depending on where you live, and how deep you need to sink them (at least 2' in the ground is best,) you can get 8' ones and get 6' of head room inside. If you can, go ahead and get them 10' (so one day, you can add a tin roof, higher on one side. You will need stringers (boards running between the posts) on the top and at the ground, and you will have to have at least one support in the middle of the long walls. I used 2x6s for this, but you could get away with 2x4s if you had to. 2x6s on the bottom and 2x4s on the top would be a good compromise. With this plan, you can improve on the structure in the future and even wall it in and add a huge run later.
I agree with Chickadoodles on the 2"x4" dog wire, but that can come later if necessary, it is sold in a 100' roll and is pricey. I found that 7/16" OSB is about the cheapest thing you can use on a coop. You could make the back wall and both side walls solid up 4' with 4 sheets of OSB and paint it with cheap "oops" paint. The last time I bought it, I paid less than $8 a sheet for it. Then from the OSB up, you can use the dog wire or the chicken wire you have, but I strongly recommend that you double that chicken wire for security AND that you also cover the roof with it if you have enough. Plan to replace that chicken wire as soon as you can afford to, it's not going to keep the preds out for long.
Now, on the front, I would make one walk-in door, The easiest way is to find an old door and hang it on the 4x4 on the hinge side and put a 2x4 on the knob side for it to close against. You could run a rope through the knob hole and around the board to secure it! Inside, you can temporarily partition off an end, or build a cage for the two birds until you can integrate the groups. You could even put 2 pieces of OSB on the bottom of the front wall, you would have a scrap 4x(width of door) to use on nest boxes or a pop door!
As for hardware cloth, I bought a 2x100 roll online and paid less than $45 including shipping. Shop around.
Materials - all wood is PT, prices are from Lowe's online.
4 - 4x4x8 @ 7.17 = 28.68 (corners only, for another 14.34, you could use them in the center of the long walls)
6 - 2x6x8 @ 8.27 = 49.62 (bottom stringers)
16 - 2x4x8 @ 3.77 = 60.32 (7 top stringers, 2 ceiling supports, 5 long wall supports, 2 door frame)
4 - 4x8 7/16 OSB @ 7.17 = 28.68
4 - #50 bags of Quikrete for setting posts @ 2.42 = 9.68
This is a total of 176.98, but you would have a solid structure to which you could add a strong roof, or even a floor and use it as a shop or storage building if you ever get out of chickens. I know there are still things you will need (screws, tarp, wire, etc.) but if you have to scrounge and make do on those things, they can be replaced or reinforced later as you can afford it, and the whole thing will only get better.
Many people build something "junky" for their first coop (I did) and end up spending much more to do it right the second time. I wish (there I go again) that I had saved the money I spent on the first one and just built the second, nice one, to begin with.