Amount of classes for a semantic segmentation neural network
It's hard to tell whether you are planning on doing image classification or object recognition/detection.

If you're doing image classification, you need each image to have only a single object, and the object needs to be in approximately the same position every time.  You train a neural network to output the label (which object it is).

If you're doing object recognition/detection, in the training set it's not enough to have a label for each image.  Rather, for each image you should have a list of the objects (labels) and the bounding box of each.  Then, you train a neural network to output the label (which object) and location (bounding box) of the object.  That requires a different architecture than object classification.

I suggest you spend some time reading up on object recognition/detection and how to do that with a neural network, since it sounds like you might not be familiar with that yet.

As far as the background class:  I'm assuming you mean a "none-of-the-above" class.  If you are doing image classification and you expect to have some images that aren't of any of the 12 labels, then yes, you should have a 13th label for "none of the above" (i.e., background), and your training set should contain examples of all 13 classes.  If you're doing object recognition/detection, that isn't necessary; instead, typically the training set needs to have realistic images that contain zero, one, or more objects, though that might depend on the specific architecture and approach for object recognition/detection you are using.