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Do you have the Snow Leopard installation discs ?
That is strange. Time Machine would not mess up anything. Not sure what happened.
Apple has replaced iPhoto with new application called Photos app. Photos app does support your old iPhoto library. So you will be able to open Photos and see all the photos that were in iPhoto. The new Mac will come with Photos instead of iPhoto so it is better to start using Photos so you won't have any issues with migrating the library when you get it.
The only iPhoto that support Mac OS X 10.10 or 10.11 is iPhoto 9.6.1. It seems like you didn't upgrade to latest iPhoto prior to the OS X upgrade you did. That it is why it is not working now. It is not your fault, it is Apple who did this. Apple has also removed iPhoto from App Store. So you will not be able to get it from App Store. But if you get access to another Mac with iPhoto 9.6.1, then you can copy it to your Mac and run it.
When you get a new Mac, you can easily transfer everything from your old mini to new Mac using Migration Assistant --> https://support.apple.com/en-in/HT204350
Sorry for the delayed response.
Yes. A complete restore from Time Machine will revert it back to old OS and Word/Excel along with the files backed up then. To do that, you have to boot into OS X recovery and select Restore From Time Machine from here. To boot into OS X recovery, restart and immediately hold down Command and R keys for 20 seconds. Or press and hold down Option key immediately after restarting your Mac and select Recovery HD.
But I'd recommend to backup or transfer everything from your old Mini or new Mini before doing the restore from Time Machine. So if anything goes wrong, you'd have the backup. Office 2014 comes with Outlook instead of Entourage. You can use it for email.
Yes, we can continue anytime you want.