Unless you set up Time Machine to back up to an external drive, it's not doing anything. That is a whole other topic! But if you decide to try replacing the SSD, the steps I would take would be:
1. Get an external drive for TimeMachine backups.
2. Turn on TM and backup the system. (That external will now be handy once we sort out the problem and you can use it for backing up the new system.)
3. Reboot to the Recovery Partition (see
this article for how.)
4. Do a complete reinstall of the OS from there.
5. Reboot normally and see if the problem has been fixed. If so, no need for a new SSD at all, it was software. If it boots fine, restore your files using TM to restore the files and folders.
6. If it still has problems, you can go ahead and order the replacement SSD. Once it's arrived and you are ready to swap, do another TM backup, replace the drive, install the OS from the internet, (see the same article I linked earlier) then update to Yosemite and do the restore from the TM drive of your files.
As you said earlier, it's not a quick process to get all this done. If you need a machine quickly and have the funds, a new machine will be ready for you without as much investment in time. I'd estimate that if you do those steps, you're looking at 10 hours or more. Download and reinstall of the OS in step 4, 5 and 6 will be a couple of hours each, if not more, depending on your Internet speeds.
One more thought: If you find an Apple store is relatively convenient in Canada, you could take it there to the Genius bar to see if they can diagnose and/or repair it for you. Customs may or may not be an issue, but I've carried my laptop through Customs before with no trouble.