Even after resets get "The backup disk image "/Volumes/FreeAgent GoFlex Drive/MacBeth (2) 1.sparsebundle" could not be created (error 22)."

Question

HELP!  "The backup disk image “/Volumes/FreeAgent GoFlex Drive/MacBeth (2) 1.sparsebundle” could not be created (error 22)."
TC estimates over 750GB for full backup.
Late 2010 MBA is connected wirelessly to
802.11n (1st gen) TC that is connected via ethernet hub built into house to motorola router using charter.com for internet.
I have had same problem for over 150 days and multiple reboots, restarts in proper order, firmware updates, unplugs with 5min plus gaps, bypassing the tc and using external 500gb WD mybook via usb to time capsule, replacement of the WD mybook to new 2TB Free Agent GoFlex drive, etc, etc, etc!  Note: I originally named my MBA hard drive MacBeth, but at some point months ago, it said backup already existed and system automatically created MacBeth (2), worked for a while, then stopped again, despite trying nearly daily.
I gave up and started using Carbon Copy Clone daily for just the 128gb on internal SSD of my late 2011 11" MacBook Air, but need to get past these time capsule errors to back up the external disk!  Due to lack of space in MBA, I had long ago transferred both itunes and iphoto and all imovie/fcpx files to a 1tb external disk (WD Passport) that I keep in my bag with MBA.  It is partitioned with 150gb for cloning the MBA, and over 750gb for media/itunes/iphoto/etc.
Also, I have endured months of "your hard drive is full" messages and system lockups, applying the solution given to me by genious bar (deleting all aol.com outboxes/recovered messages/spam/trash under library/mail/V2/Mailboxes folders, empty trash, restart....watch recovered hard drive space mysteriously shrink next couple days to zero.....repeat)
Due to lack of space, I had not been able to do javascript and other system updates.  Yesterday, I noticed even safari bookmark bar had mysteriously emptied (not disappeared, I can hide and show the space where the bar used to be - it is just blank area with no data except for eyelass-book-topsites icons), so I transferred a few large quicktime movie files recorded with isight camera to the external disk, and freed up 20 gbs for the first time!
This got my hopes up yesterday because after I completed all system/software updates, suddenly the bookmark bar showed up and I was able to start the time capsule backup! It took overnight preparing for backup, but finally started and progressed to 12gb of the 100gb+ needed (i guess the update where it left off months ago for the estimated 750gb for the full backup?), then I put system to sleep and took it with me (thus leaving the wireless time capsule network as I do daily fully expecting it to continue when I returned home in the evening.)  But alas, upon return, back to the "Time Machine could not complete backup, The backup disk image ".....sparsebundle" could not be created (error 22)"  Ugh!  More reboots, unplugs, 5min gaps....still Ugh!
So the problem continues...please help!  Here are my settings:
About this mac:
Mac OS X V. 10.7.5
processor 1.6 ghz intel core 2 duo, memory 4 4g 1067 MHz DDR3
software up to date!
Airport Utility:
V 6.2 (620.33)
both internet globe and TC show green dots in graphic, however internet occasionally shows yellow then goes to green, but still gets error 22.  Suddenly TC showed explanation mark and says lost connection and asked if I wanted to 'forget', gave same errors, then mysteriously went back to green, but still got error 22!.
Internet:
connection: connected, displays router address, 2 dns server, and domain name: charter.com
Time Capsule:
shows network, ip address, serial number, version 7.6.3, status: (green dot) Setup over WAN,
wireless clients: phones, macbooks (including this MBA), printers, wii
If you need other settings to help me, just ask!  I have read a few of the other solutions in forums, but trying not to erase disks, especially since it worked for a minute yesterday!

Answer 1

To step sideways.. why not just use CCC to backup the 750GB of the hard disk to the TC.
You need to understand this is going to be a super slow process and each time you stop it.. it is going to give issues. CCC will at least do a more reliable job of it than TM.
I would also say.. this is going to be tough without ethernet. You need to understand the backup speed over wireless can be 10GB per hour.. that makes 75hrs to backup the disk. The air needs to run non-stop doing that.. without you touching it.
I would get another external drive.. plug that into the MBA and if you really want to use TM.. make a backup one to the other.. but CCC will again do a better job and work faster. It will be much faster than over wireless.
I am trying to think of ways around the issue of just speed.. but plainly you have lots of other issues.. being the simple unreliablity of the TM to identify the TC.
If you want to make this work.. give yourself a weekend.
Load 5.6 utility into the MBA..
http://support.apple.com/kb/DL1482
Take the TC back to 7.5.2 firmware. It is a really old version and latest firmware does it no good at all.
Wipe its hard disk using erase and start over.. give it new names for everything.
So call it TC
Set the wireless to 5ghz and name it TC5ghz
Use WPA2 Personal security. Use a 10 character passkey non dictionary.
At 5ghz you should get 2x the speed as at 2.4ghz but the air will need to be right next to the TC..
(alternatively buy the USB to ethernet cord apple have available.. it is $30 in the apple store and do it over pseudoethernet. http://store.apple.com/au/product/MC704ZM/A/apple-usb-ethernet-adapter That is the wrong store but you get the idea. )
WIth a fully clean system.. now do a full reset of TM.
See A4 http://pondini.org/TM/Troubleshooting.html
Direct the TM to back to the TC.. and make sure the external hard disk is not excluded.
Of course the TC has to be 1TB version to fit that much data.. goes without saying right.
Even over pseudo - ethernet this will take many hours.. and you basically have to leave it to work.. do not touch it.. make sure the MBA cannot sleep though.. turn off sleep.. beforehand.