Use the 'exclude invoice lines' option on the late fee policy to exclude specific accounts (Xero) or items (QuickBooks) from the late fee calculation.

This would allow you to have an invoice with, for example, two line items: Rent $500 and Admin fee $20 that is only partially charged a late fee. If you excluded rent from the policy then a 10% late fee would only apply to the admin fee line and so the late fee would be $2, not $52.

Can an entire invoice be skipped?

Yes - If an invoice is completely excluded, i.e if all the line items are excluded, then it will never have the policy apply.

What if an invoice has been partly paid?

Because a payment is applied to a whole invoice and not specific line items, Paidnice will take the total of the excluded lines away from the balance owing. If the there's no amount remaining then the invoice is not applicable and no late fee will be charged.