merging photos automatically and getting the background to match it.

To extend the background colour, you can use Content Aware.

As below, I selected the white area above and below the characters, then went to Edit > Fill and in the window, selected Content-Aware, then click OK. The result is in the second image below.

For more on Content Aware, see
Content-Aware Fill workspace in Photoshop​

As mentioned by others, the lighting may be different between pictures so even if you extend colours as above, you could get a flicker when making the video. That could be solved in other ways. One option is to mask out the background.

I would add, that as an "absolute novice", this is not the best project to start your Photoshop journey on!

There might be something useful in this: How to Create a Stopmotion Movie - Photoshop Tutorial - Pxleyes.com

To get rid of the white background I tried using the rectangle tool - to draw a background in the same color as my photos, but the colors do not match up (even if I use the eyedropper tool). I have no idea how to actually do this, so I need expert advice in how to fully utilise this powerful program.

Your images’ backgrounds are not »flat«, they probably have noise and show some of the structure of the original material and they are apparently not lit perfectly evenly but are lighter at the top edge than at the bottom edge.

Maybe you need to mask the subjects, how easily that can be automated is hard to tell just yet, can you provide a few of the images?

As it will be a stop motion, each photo needs to be the exact same size every time I combine it. Is there an easier way to do this, instead of dragging the photos in and adjusting them manually one by one (I have 180 photos which needs to align seamlessly when played as a video)?

You could try

File > Scripts > Load Files Into Stack (to get the whole bunch into one file as Layers)

Edit > Auto-Align Layers (to align the selected Layers, though this may not work out perfectly)