Sort and filter contacts by number of open invoices, and by dollar amount owed
I would like to be able to sort (and filter) the contacts list by the dollar amount owed, and especially by the number of open invoices for each member.
This would help me when sending out reminders to members that have fallen behind on their payments.

No activity for long time. You can use advanced search with Balance due search criteria and set some threshold to it
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You can use "Attention required" quick filter (dropdown in Contacts / List), it'll show every one with problems, including outstanding balance. But this won't solve your problem fully, I agree.