I hooked up the ethernet cable to the iMac and turned the wifi off and the wifi still shuts down and no backup takes place.
I am a bit lost here..
If you turned the wifi off, I am not sure what you mean by wifi still shuts down. Do you mean the wifi in the TC itself shuts off?
There is no doubt Mavericks has made life extra hard. Did this work ok on 10.9.3 and went to pot on the latest update??
My suggestions will be to redo the setup.
Start with factory reset of the TC.
Make sure ipv6 is set in the computer to link-local only.
Give the TC all new names, short, no spaces and pure alphanumeric.
See if that fixes it.
The factory reset will cause no harm to the existing backups.. but TM will need to be reset and will say it is going to start over.. but will actually find the old backup and continue.
If you still have issues.. manually mount the TC in finder.
Use Go, Connect to server.
AFP://TCname.local (where TCname is your new short name and local is the default domain).
or even better if the TC is router.
AFP://10.0.1.1 (replace with the actual IP but that is default).
There are lots of details for the above in other posts I have done. eg
https://discussions.apple.com/thread/5800974?tstart=0
Airport Utility shows "unexpected error - please try again" when connecting to Time Capsule