In regards to the original post question about separating roosters, I bet that was for straight-run numbers, which usually are more around 50/50%, than sexed chickens with the occasional possible male. So one (or two) in your 10 will probably be manageable. (Side note- I did start out with 4 EEs from the local feed store as my first flock, and there's nothing wrong with getting pet quality birds from a feed store if there are no other easy options where you live. One of those 3 did turn out male though, so that 90% correct sexed claim didn't work for me. lol)
Like a few have said, the internet is full of people who either don't think, don't care, or don't comprehend how their words are taken when talking to newbies. I've experienced it on other forums with other hobbies, and being made to feel stupid when I've done my research makes me feel awful. But you just learn to ignore the unhelpful responses and look for the knowledgeable helpers. There are many helpers here.
My dad built my chicken house for my 4 chicks. It is raised off the ground and 4'x4', so 16 sq ft for 4 chickens, as per standards. That sits inside a 10x10 chainlink dog pen that has been fortified for protection. Then I later added 3 rescue bantam chickens, so I still have appropriate run space but my house is technically "too small." But honestly, they usually sleep on top of it anyway, and even when I put them inside for bad weather they all fit just fine for the night. I cannot free-range so they're always in that pen, but have plenty of room on the ground plus areas to go up/sideways and toys for entertainment.
I think by looking at the pics of the coop you linked that you may be able to make it work for the 10 chickens you have ordered. It looks to me like the house is 4'x4', based on the outer dimensions they give and my personal 4x4 house. Especially if you are planning on something bigger in the short future, I think 3 chickens will fit on each perch. (Check the size of those perches. Mine sit on 2x4s with the 4" side flat, so they have plenty of space to rest on. You might want to upgrade yours if they are 2" or less wide. Also, probably remove 2 of those nest box dividers, just leave the middle one to make 2 larger nest areas. You don't need 4, and those look small.) I'm not saying you'll have only 9 chickens, but I know that chicken math also includes subtraction, as something could happen to one or one may be a rooster you eventually rehome. I'd bet 10 could fit in there if they had to. It's not regulation size of course, but...
... what I'm actually going to recommend to make things work, (instead of trying to cram the chickens in there nightly,) is to make sure that the actual pen is secure then not lock them up at night and add branches/perches/roosts inside the pen itself. My whole coop is literally against my house under my 2 story deck for protection, so they are covered and secure, and unless it's wet AND freezing, I don't lock them up at night. They can get on top of their house and love to roost up there, as chickens will go to the highest point they can to sleep. They are protected from the elements, happy, and healthy, and do have the option of the house if they want it. They still go in to use the nest boxes and will play around in there, but would rather sleep outside, up top. Your pen being 13x13 gives you plenty of room to add roosting places for them that would be protected with that top canopy, and I think your chickens would be happy to have the choice to sleep outside or go inside the coop if they wanted. As long as predators can't get in, there's no reason they couldn't do either and you sleep sound at night. Especially if you are going to free range them on your property during the day then lock them in the coop at night, the whole coop (pen and house) then becomes their secure sleeping area. I know we all want to baby our pets, and I used to make sure they were tucked in every night for safe keeping... but I've found that mine are pretty self sufficient and are much happier sleeping where they want instead of me going out there and trying to catch/cram them into the house.
So don't stress. Get your chickens, get your coop, use the time while they're growing in their brooder to get the coop set up with some extra perches/roosts for them to play/sleep on. Spend this first year getting some first-hand chicken knowledge so that you know what you want/need when you expand the next year.