I can create a query to select only pads and a query to select only components who are not type "No BOM", but I can't seem to find a way to put these together to create a design rule that only selects pads from components that are not type "No BOM". Any ideas?
You can combine any series of rules to narrow down groups by using the keywords "AND", "AND NOT", "NOT", "OR", "OR NOT".
Like so: RuleOne AND RuleTwo --> Both need be true
If they are complex rules, put them in brackets: (RuleOne) AND (RuleTwo)
To be sure they get handled right.
You can test rules easily in the PCB Filter dialog (from the PCB button in the bottom right of your work area when in PCB mode and typing them there and see what gets left coloured. Everything that doesn't belong to things that comply with the filter you type will grey out, until you click "clear".
Loads of information is given in the Altium Design Rule Reference, but as is usual with the Altium Tech Docs, it's not the easiest to read.