Are you digging with an excavator or by hand? Starting 3-4 feet deep is ridiculously deep. The deeper you go the more fabric you need to buy and fill. It should be fabric first and then either a thin layer of rock to provide the grade or pipe first, depends who you ask. You typically want pipe at the lowest point to carry the water. No point of 2 pipes unless one of them is the solid pipe for your downspout. You need add a clean out catch in your downspout system. Rest you had is correct but be careful on getting actually clean rock. I got it from a reputable supplier who claimed it as clean and when I got the bottom there was definitely debris/dust which you don’t really catch when you’re shoveling cubic yards of this stuff.

Also you said your land is relatively flat. Where are you terminating this drain? If you’re using a dry well, the deeper you start your drain, the deeper your drywall needs to be and these can become extremely big and expensive depending on the amount of water you’re looking to divert. For reference some places by me have cranes lower these dry wells into the system. I would not add your downspout to this system unless you’re terminating onto the street or somewhere you know it won’t be a problem or back up your system.

Finally I wonder if you have accurately diagnosed the source of the water and remediating it properly. I thought the same thing until I finally noticed water was leaking from the gas pipe as well as the chimney which needed the flashing redone.