If I'm using PBKDF2 to stretch a password into a suitable key for use in encryption with AES, should I be using a random salt (for PBKDF2, not the AES IV)?
Absolutely! It's even right there in the PKCS #5 v2.0
By using a secure salt, password
You must do this, as otherwise two users using the same password twice will generate the same key. The effort of using a salt is almost none, yet increases your security.
Please refrain from rolling your own there is a standard and you should use it!
It's healthy to question the "whys" from a learning perspective but playing with crypto and trying to cut corners can be dangerous if you are not an experienced cryptographer.