purchases and other expense SPs

See the response to your other post of today about the assertions for transactions and balances not being the same. However, where they are the same, yes it will be true (must be true!) that a test for completeness of payables must also test purchases for completeness.

Yes - this must in essence be true if you are talking about the "opposite side" of the entry. So if the recording of credit sales transactions is complete, the recording of trade receivables must also be complete with regards to the recording of credit sales in trade receivables.