Using Airport as External HDD using usb cable

Could I use a male to male usb cable to connect the two, then be able to read and write files on the Airport's HDD?

Sorry, but no, since the USB port on the Time Capsule is there only to allow an additional hard drive to be connected to the Time Capsule for more storage, or to connect a compatible USB printer.

If not, then please tell me how to set it up through my network and not use it as a router

Connect an Ethernet cable from one of the LAN <--> ports on your main router to the WAN "O" port on the Time Capsule, set up the Time Capsule in Bridge Mode (which turns off the routing function of the Time Capsule) and then turn off the wireless function on the Time Capsule (to prevent the Time Capsule wireless from interfering with your network's wireless service.) Apple's setup "wizard" will do most of the work for you automatically. You will then use AirPort Utility to manually turn off the wireless on the Time Capsule as a final step.

As you might imagine, Apple will frown upon this type of setup, because they expect you to use the Time Capsule wireless.

Also, i did 2 factory resets and it would not let me "set up as hard drive" as some websites suggest

Not sure what websites you are referring to or what "it" might be, but the instructions above will do what you want.....to set up the Time Capsule as a network hard drive. If you are not familiar with Apple's set up "wizard", post back for more details. Here, I am assuming that your Mac is running a current or recent version of the operating system.

It would really help if you could provide the make and model number of your router, and what operating system the MacBook Air is running at the present time.

And I have a macbook air so I can't connect it via ethernet to my computer

The setup "wizard" assumes that you will be using WiFi to set up the Time Capsule, so an Ethernet connection is not required on your Mac.

If you temporarily power off the Time Capsule, there is still another device there that is providing a wireless signal.....according to the information in your post. That device is your router.....to which the Time Capsule will connect.

You might call this device your "modem", but if it is providing a wireless signal now, it is a modem/router or gateway device.

We need the make and model number of this device.

Once we have that information, we can provide step by step instructions to do what you want, if you need them. Please advise.

Not sure what you would hope to gain by connecting the "old" router to the "new" router unless you plan to locate the old router at least a few rooms away from the new router and connect the two devices together using a permanent, wired Ethernet cable connection. This would allow the old router to provide additional wireless signal coverage in the area in which it is located.

But, to do this, your old router will need to have a feature that allows it to operate in Bridge Mode, so that the old router will be on the same network as the new router. If the old router does not have a Bridge Mode option, then the old router will be producing a different network, and devices on the old network will not be able to communicate with devices on the new network. You will also introduce an error known as Double NAT, which may or may not create some other issues, like slow browsing, slow connections, etc.

It won't hurt to connect the old Time Capsule to the new Time Capsule using an Ethernet cable, but you would not want to have the old Time Capsule providing a wireless signal unless it was going to be located a few rooms away from the new Time Capsule.

Where will the old Time Capsule be located?