This is a "sticky" situation, but I'll try to offer some answers.
First, I'm not even sure if one can boot from such a folder. But, if you could, it would show up in System Preferences, under Startup Disk.
Secondly, in order to do a fresh installation of OS 10.11.1 (I assume you have the latest version), and given that you do not have a backup, there should be a Recovery HD partition on your new drive that was created when the shop did the El Capitan installation. But, the
first thing you should do is to make a backup of the "partially problematic" internal drive, as you can at least run your machine with that. Time Machine is OK for that, but a cloning program like either SuperDuper! or Carbon Copy Cloner is
MUCH, MUCH better, as either of them will make a bootable backup. Next, you would need to boot to the Recovery HD partition. This link explains what is on that partition, along with how you boot to it:
https://support.apple.com/en-us/HT201314
As you can see, you can then 1) use Disk Utility to Erase and Format the internal drive, and 2) install El Capitan (hopefully it OS 10.11.1). You should then be able to re-boot your machine from the internal drive.
I believe Migration Assistant could "migrate"/copy stuff from that folder, no matter where it is. And, you can pick and choose what it should "migrate", but I don't know how "detailed" you can get with that, as I have never done it.
Once you get to a successful state, I
strongly suggest you start making a backup as often as necessary, so that you will not risk losing those 7000 photos (and any other valuable stuff)!