I've been trying to figure this out for a while and, clearly, have had no real luck.

I've looked into Anach's Newspaper Tweak mod (which adds a $5 paper delivery cost) but I can't understand what or how she did it.

Which brings me to... I want to make diapers and bottles cost money, like they do in real life. How would one go about adding a cost to those?

Do you want it to occur at a certain time? You can use the "ModifyFamilyFunds" command to write a script that does that ... you just need to decide when you want it to happen.