When in the Airport Utility / Time Capsule I can click Edit and get some options,
That is great start.
in the Wireless tab I put in the router name and password,
Normally we will set this to wireless off. Unless your main router is a lot older than A1409 which is 7-10 years old. Or you have a very long ethernet cable and the computer is right next to the old TC.. normally just turn wireless to Off.
in Network tab I have Router Mode in Off (Bridge Mode) selected
Correct.
In the Disk tab, am I able to look at what's on the drive by selecting Archive Disk?
No.. archive disk will copy the entire contents of the TC disk to a USB drive plugged into the TC USB port.
To see the contents of the disk you need to mount the TC disk in Finder on the computer.
If there is any content you want to keep then copy it to another location or do use the Archive function.. It might be well worth doing an erase of the drive and a Zero Out Data full low level format to prepare for usage if you expect it to work for another few years.. remember this disk has had a hard life and is now also 7-10 years old. They were cheap WD green mostly. Good drives but not for LONG-long term use. The low level format will take half to one day.. and will check every sector is readable. *** This removes all Data from the drive*** So if you need to keep some or most of it don't erase.. at least don't erase before you archive.
What OS is the computer running now?
What router are you plugging the TC into?
How would I proceed to turn it into a 'usable' hard drive, through connected or better yet wifi?
If wifi here means via the Main Router wifi.. that is fine.. it will connect over wifi to Main Router and then to the TC over ethernet.. which is connected via the main router... Leave wifi off on the TC.. it is only going to cause issues.. unless as stated your setup is Main Router and TC are widely separated and speed to Main Router over wifi is poor.
There are a couple of other things that can help.
1) Use short names, no spaces and pure alphanumeric for the TC name.. e.g. tcgen4
Same rule but longer for the password.
2) Use static IP. This is IMPORTANT. People often have issues because they do not follow this.. if it does not make sense post again with more info about your main router (its home IP and dhcp range and we can give more detailed instructions).
Just select static on the TC.. you must manually fill in DNS.. use either same as router or public DNS like 8.8.8.8
Even better if you apply this via DHCP reservation in the main router.. but not essential.
N.B. Yours will not look exactly like mine.. illustration only.. and you can move the IP to a more sensible number.. easy to remember as long as it is not already allocated on the network.
3) Use link-local only for IPv6
This is under internet options.
Your computer can still use IPv6 to internet but will use IPv4 to the TC.
Knowing the Address helps also as you can mount the TC in Finder before you try setting it up in Time Machine.
If that fails after a few days due to poor handling of bonjour by the main router you can use IP address instead. Post again for more details if you need it.