Undeposited Funds is used when including payments because that's where your payments are recorded.  Picking the Undeposited Funds account directly, which technically allowed, won't work properly to deposit the payments.

It's a better practice to create the deposits in QuickBooks as you make them, including the payments in them at that point. After doing that, when you download from the bank they'll already be in your bank register.

When using invoices, neither the payment nor the deposit uses any P&L accounts, like income accounts. The income is recorded on the sale - the invoice. If you report on a cash basis, then the income will usually use the payment's date (if the payment follows the invoice by date). Using either accrual or cash basis reporting, the deposit has nothing to do with income.