How many hatching eggs will you have? I know not all may hatch but you have to plan as if they all will. That's going to determine how big to build the brooder. It needs to be big enough for them as they grow. You do not want them to outgrow their brooder. If they get crowded their behaviors can turn dangerous. How many might you have and how old will they be when you are ready to move them out of the brooder?how heavy that will be to get it in and out of the house etc.
To avoid all of the issues with it being heavy or having to move it through doors, build it in your heated garage. Start them in the brooder in there. You could probably even hatch in that garage if you wanted too. I don't know how warm you plan to keep it but I'd think that is your simplest option. It may avoid your cat issues.
My brooder is in the coop. I put them in there straight from the incubator or post office. I don't know what your coldest temperatures could be while they are out there, my coldest has been about 10 Fahrenheit (-12 C). I use a heat lamp to keep one end of my brooder warm enough in the coldest temperatures and make my brooder big enough and with enough ventilation so the far end cools off enough in the warmest temperatures. I keep my water in the warmed area so it stays thawed. My biggest issue with brooding outside is the temperature swings. Mine can go from below freezing to the 70's F (21 C) in less than a day. I don't want to be out there constantly making adjustments especially at night.I do have a coop already actually, but there are 9 full grown hens in there so I’m not sure how easy that integration process will be.
I agree with @azygous , if they grow up with the flock integration is usually a lot easier.
How big is your coop (in feet or meters). Will you have enough room for all of your new chickens? Integration typically takes more room than if they are already integrated.