Both of my drives (HDD & SSD) are installed, right?

Yes.

Did I somehow manage to load the OS on the HDD and not the SSD?

The OS itself is on the SSD, but the boot partition which stores files used by the Windows bootloader is being stored on the HDD.

That might cause your booting to be slightly slower than if it were on the SSD, but honestly the speed difference is likely to be quite small, to the point where you probably would not notice it at all.

Is that reversible, or will I have to start all over again?

There are ways to move the boot partition, and even to delete the boot partition and have Windows store all of the boot files together with the OS on one partition, but honestly doing so is not the simplest thing to do and will probably be more trouble than it's worth.

If you do care about moving it then I'd recommend googling "Move boot partition Windows 7" or just reinstalling Windows without having the HDD connected.

Can you do that with the "builder's edition"? The box says you can only load it once

The license only gives you the right to install Windows to one machine, reinstalling Windows on the same machine is fine.

I found some information that states that I can view both my drives if I assign the drives a letter, but my HDD is already assigned, under C?

C: is assigned to your Windows install which is located on your SSD.

Drive letters, contrary to their name, are not actually assigned to individual drives, they are assigned to partitions on the drive. You can have multiple partitions on a single drive.

Your SSD has one partition which fills the drive, and that has the drive letter C:, Your HDD has only one parition which stores Windows' boot files, this parition is only 100MB, the rest of the space on the drive is not partitioned, in order to use that space you have to create a partition that uses that free space.

Doing so is easy, just right click on the area that says "Unallocated" and select "New Simple Volume..." and follow the wizard that pops-up.

After creating the partition the drive should already be assigned a drive letter (Most likely D:), that drive letter will be where you store stuff when you want to save it to the HDD.

If you have any further questions, problems or found any part of this comment confusing, then feel free to say so and I will gladly try to help you out.