Payments do show on the invoice. After you have "Applied" the payment, just regenerate the invoice and you will see it and the adjusted total.
We do have a "Discount" option on the Invoices as well. In the Extra's section, just choose Discount and set the Discount Type (percentage or amount) and then enter the Amount. One important note is that discounts are taken before Taxes, so if you are discounting your invoice, that is a good way to do it.
As far as payments go, you can Apply a payment to one or more invoices. So it could pay two, three or more different invoices.