aA 49 sq foot coop is not big enough for 13 birds. You are talking just over 3sq feet per bird. If you have to keep the birds locked up for longer periods of time you will have injuries. Birds peckign at each other, fighting for space. You want to have a minimum of 4 sq feet of floor space per bird, with a foot of roost space per bird. This floor space does not include the space for nest boxes or feed stations. Nest boxes and feed stations really will cut into that 49 sq feet. You want at minimum 10 sq feet per bird in the run. You'll just need to settle for fewer birds. Say 10 hens and 1 roo.
NOW, that being said... if you are going to have FREE RANGING birds, you can get away with the smaller coop for them all. If you are not, you really need to have a much more extensive run. 130sq foot minimum, that'd be a 10x13 foot run... Personally, I'd double that length. Say a 13x20. Trust me, you'll be happier for it, since the birds will be happier for it. Yes bury the whole perimeter of the fence. By having it be long and skinny, you can roof it easier with PVC and tarps, to create shade, or even dry area. My run is 12x28. PVC is makes the roof frame, and a tarp. The sides are plastic covered for the winter. I will leave the tarped roof on for the summer as well, to keep a dry area that is always shaded. ALSO, able to make separate pens and coops with in that space if needed. I have 4 coops with in that space, all up off the ground so I can crawl under as needed.. I used to let my birds free range all day, and lately have had to lock them up more often, the hawks have been horrible on them... SO, I am happy that I have that much run for them. They've all got enough space. They aren't happy being locked up, but they aren't dead either.
So, all in all, if you are going to have a coop that small for 13 birds, then you really need to make the biggest possible run that is predator safe. I even have roosts in the run, and a lot of the birds prefer to sleep on those instead of in the coops. I only allow that though because I am pretty confident that nothing is going to get in... You need to be confident nothing can get it...