Someone at the Apple Store may be able to see if the user account in the old computer can be
accessed (Target Disk Mode) and some of those old Libraries such as iPhoto, may transfer.
There is a migration assistant that could try & move files and content from an external drive or
a backup on an external; or perhaps from a computer that is booted into Target Disk Mode
where it may be seen by another computer as an external storage drive.
Was the Snow Leopard install-restore DVD used to see if an archive and install could be done
and a new system put into the old computer, in a way that does not erase the content? If the
original hard drive is worn or failing, that could contribute to issues and faulty results.
If an independent Apple authorized service provider is an option, (the computer may be too old
for Apple Store to restore it; they may be able to transfer the old Libraries from it to a new one)
the old and tired Mac may have some life left in it. Also, it may be the hard drive is too full of
user accumulated content and the Yosemite was unable to budge.
A slow internet connection may also have been a partial reason for problems, from trying to
upgrade from Snow Leopard 10.6.8 to Yosemite 10.10... Could be the computer didn't like it.
Without exact system specifications, and that means hardware build version + etc, it is hard to
say if the old Mac was ready or able to accept an upgrade. Some are happiest with 10.6.8.
A new hard disk drive, installed in the Mac, by use of Snow Leopard install DVD so it would be
formatted correctly and a new system installed properly, is one way to get the computer to run.
And the old hard disk drive could be put into an external drive housing, and attempt to transfer
or migrate any old account data, tried with Migration assistant; in the absence of time machine.
Then the old HDD would not be overwritten by a new OS X. It may have seen overwriting by
the Yosemite installer; or it may have failed if the hard drive was weak, prior to full installation.
A non-bootable drive may still be accessible by a healthy Mac. Also, a universal USB drive kit
could be used to attach a bare drive to a working computer to try & use a hard drive...
A 'univeral drive adapter' of some type may be located at a good computer shop, a better one
is this http://eshop.macsales.com/item/NewerTech/U3NVSPATA/ - newertech USB adapter.
Best of luck in this matter! 🙂