What is the connections configuration that I should implement to output a usb-C and usb-A 3.0 from a usb-C input? The input will be from a docking station capable of supplying 20V to charge my laptop. It can also be used with a usb-C to usb-A adapter to only use it as a docking station without supplying power to charge the laptop. However, since the usb-C port to my laptop doesn't work properly, it can only be used as a charger. Therefore I would like to design a PCB that outputs a usb-A to use as a docking station and a usb-C to use as a 20V supply to my laptop. So far I figured out the connections if I had to achieve this separately. But I'm not sure how I can achieve this combined when CC1, CC2 and VBus are common to both outputs.
I'll add to the pile of comments suggesting that this is outside the realm of something an electronics hobbyist could build. I believe the proper solution is a different dock.
I may be a bit confused on the dock description. Does the dock have a captive USB-C cable? Captive as in it has a short cable with a male USB-C jack on the end? And the dock came with an adapter to plug this USB-C cable into a USB-A port on the computer? An adapter that allows a USB-C captive cable to plug into a USB-A port violates the USB specs, anyone that included such an adapter with their dock is not taking good engineering practices to heart. I have a suspicion that the reason the USB-C port on your laptop is not working properly is because the dock damaged it. By connecting a poorly designed dock to another USB port on your laptop is just asking for another USB port to be damaged.
Perhaps I assume too much and the dock is fine. What remains is that an adapter with a female USB-C port and male USB-A jack violates the USB specs. This is an adapter that should not exist as the people that wrote the USB spec explicitly state that such an adapter should not exist. There is no knowing how such an adapter is wired because the spec gives no guidance on how to wire it. The way it should function is ambiguous, and there's several ways in which using this kind of adapter can damage anything connected to it.
Another thing that I'm a bit confused about is why you don't simply plug the dock into the USB-A port and the laptop charger into the USB-C port. If by chance you were able to build a functional adapter as you proposed then you'd still end up plugging two plugs into your laptop when it's set down to be connected to the dock. Why bother building this adapter when it's so easy to just plug the dock in one port and the laptop charger into the other?