If I upgrade to mesh router, can I continue to use Airport Extreme as a wireless back up for multiple devices?

Will setting up the Extreme as a bridge automatically disable the wireless radios, and if not, what is the procedure to accomplish the task?

No, reconfiguring the Extreme as a bridge will NOT automatically disable the wireless radios. It will take two separate set of steps to accomplish this.

1. To reconfigure the base station as a bridge: AirPort Utility > Select the Extreme > Edit > Network tab > Router Mode = Off (Bridge Mode)
2. To disable the wireless radios: Wireless tab > Network Mode = Off

Also, does the Extreme have to be connected to one of the nodes or can it be connected to the primary router?

No, the Extreme can be connected to any of the nodes, including the primary router.

I also need a mesh router with at least one USB port as I currently have two printers connected to the network, one wireless and one not.

You could still use the Extreme for this ... or review each of the mesh offerings to see which offer USB ports on their nodes ... but be aware that just having a USB port does not necessarily mean it will perform as a print server. You need to verify that as well.

FWIW, I currently replaced my Apple networking ecosystem with one provided by Ubiquiti. I use their UniFi solution as my home was pre-wired for Ethernet. They also have a mesh option as well, but Ubiquiti has a bit of a learning curve as they are not quite "plug and play," and why I didn't initially suggest it.

Most new wireless mesh system provide nodes with Ethernet port(s). In that case, you can still connect your AirPort Extreme to one of those nodes by Ethernet and continue to use it for Time Machine backups. You will also need to reconfigure the Extreme as a bridge and disable its wireless radios as they will no longer be needed in this configuration.

As far as which mesh systems to go with, will depend on, whether or not, your home is pre-wired for Ethernet. I am assuming that it is not and your goal is to have wireless connections between these nodes ... correct? If that's the case, you will want a mesh system that incorporates a dedicated wireless backhaul for the best overall wireless performance. Those would include mesh systems from: Asus HiveSpot, D-Link Covr, Linksys Velop, or Netgear Orbi. Of those, I suggest that you take a look at the last one: Netgear Orbi.

Something else to consider. As you know Apple has gotten out of the networking hardware business. Your AirPort Extreme is on "borrowed time" and you should be looking for an alternative solution for TM backups. One suggestion is to replace the Extreme with a dedicated NAS. Those provided by Synology also support TM backups.

You're very welcome! I'm glad I could of helped. The best to you as well.