I only want the paint that intersects with the black object to be visible (as well as the black object). Better yet, I would like the paint that is not covering the black object to be deleted. I have tried selecting everything, and using pathfinder > divide. The main anchor points are retained but it doesn't even look like a paint brush anymore, just a red line. Any help would be greatly appreciated!
You could create a clipping mask that will mask off the parts of the object that you don't want to have visible. In my opinion this is better than permanently deleting; you can edit it and move it around if you change your mind about the positioning of the red stripes.
Is the black shape already a vector object? If so, copy it and bring it to the front of your stack of layers (doesn't matter if it still has a black fill). This object will be called the "clipping path". Select the clipping path and the layer you want to mask (the red stripes). Right click and select "Make clipping mask". Your red stripes should now be clipped to the shape you want.
If your red stripes and black frame shape are all one layer and not a vector object then you could always create a new vector object that's the same shape as the black frame, since it's a simple shape. Just draw out two rectangles the right size and then use the pathfinder tool to subtract the middle using the shape mode "minus front" (make sure the smaller rectangle is in front when you do this then it'll minus out the middle bit and give you the frame shape). You can then use this vector object to create your mask as above.
Hope that's helpful. If you want to learn more Adobe provides a useful help page for learning about clipping masks.
P.S. I'm using CS6, the process might be slightly different on other versions of illustrator.