How do I daisy chain two time capsules as a wired router and use the hard of both as back-up?

I assuming I plug the Ethernet cable into the LAN port of the TC tower and the other end into the WAN port of the older TC. Is this correct?

Yes

What settings do I need on the TC tower and the older TC to make this work as a wired connection?

The setup "wizard" will automatically provide the correct settings to the older TC to "extend a network using Ethernet" when you start from a factory default reset condition on the old TC.

Do I need to turn off the wireless connection on the older TC?

You can, but if the older TC is located some distance from the new TC, you might want to leave the wireless on to allow an "extended" network with more wireless coverage in the area near the old TC.

Will I just need to plug in the DVR into the older TC’s USB port to use its hard drive?

The USB port on the old TC (and new TC) will only support a normal computer hard drive.....not a DVR. If you want to connect a hard drive for more recording capacity, you will likely need to use a separate USB or SATA hard drive that is connected directly to the DVR, not to the old TC.

Check the documentation on the DVR to confirm how that device should be set up.

The old Time Capsule has Gigabit Ethernet ports, so there would be no advantage at all performance wise using an Ethernet switch in place of the old Time Capsule.....which will function just like an Ethernet switch, plus give you capability to have more wireless coverage in the area near the old Time Capsule....and also provide more network storage space, if that might be needed.

Happy to help. Good luck!