Apple support can't seem to help - can you?

Question

I have a 27 inch Imac (late 2009) running Lion 10.7.5.  I am  having extreme difficulities with spotlight and time machine.  Spotlight reindexes everytime I reboot and time machine never finishes a backup. 
Thus far my Apple Care protection and a number of people at Apple have provided me with the following solutions, none of which have been successful.
A basic reinstall
Completely reformatting my time machine disk
A complete "clean" reinstall - after which I installed all programs from fresh install files and only transfered my personal documents from an external drive back onto the mac.
An attempt to Repair Home Permissions and ACL's through the Lion Recovery thing - it ran for 4 hours to no conclusion.
I also went out and purchased an entire new external harddrive to use as a time machine since I figured it could possibly be the old one (even though the apple store techs checked it and wiped it).
Oh and there is a new keyboard on the way since my wireless one wouldn't connect after the reinstall.
Any ideas as to what could be wrong.  Is it hardware? Permissions?  I've spent hours and hours (almost a full 40 hour work week at this point), on the phone or in the Apple Store and it still isn't working properly.  I'm one more phone call away from giving up and putting the thing in the recycling bin, so hopefully someone out there can help.

Answer 1

An attempt to Repair Home Permissions and ACL's through the Lion Recovery thing - it ran for 4 hours to no conclusion.
This is a problem.
Boot your Mac normally and launch Console - it is in your Utilities folder. From the View menu, select Show Log List if it is not already shown. Select system.log from the log list.
In the filter field at the upper left type mdworker. You will probably find many errors associated with Spotlight indexing. Copy a couple dozen of those entries and post them in a reply. Do the same for mds.
Time Machine will not function until Spotlight builds an index and it sounds as though it never finishes. You can force it to reindex by performing the following:
From the Apple () menu, choose System Preferences.
Click Spotlight.
Click the Privacy tab.
Drag a folder or an entire volume (your hard drive) to the list.
If prompted for confirmation, click OK.
Remove the item or volume you just added to the list by clicking it and then clicking the minus ("-") button.
Close Spotlight preferences.
Spotlight will re-index the contents of the folder or volume.
This is unlikely to help given the failed attempt to repair home permissions and ACLs though. The reason for that is a mystery. Boot OS X Recovery again, select the startup volume, and click the Repair Disk button. Reply with any errors that it reports in red.
Whatever problem is causing this may best be solved by erasing your startup disk, followed by installing OS X from the Recovery partition, followed by observing the system's behaviour prior to installing anything more, followed by correlating Spotlight indexing subsequent to installing additional software. I realize you performed a "clean reinstall" but the definition of that is subject to interpretation.