MX Merchant Ideas

Thank you for your feature request! When you are looking at the invoice in MX Merchant we show in the history section at the bottom of the invoice shows the email address the invoice was sent to and the date and time stamp as to when it was sent as well as who sent it. But it sounds like you are looking to receive a copy of the invoice to your email anytime one is sent to your customer is that correct? 
For your second concern, there are a couple ways you can be notified of a payment on an invoice. In the receipt settings, you can add your email address in the BCC Receipt To which will send you a copy of the receipt anytime a payment is made. However, if you just want to be notified when a specific type of payment is made if you go to notification settings, click the green "+Add Notification" button in the upper right. For the Type select Payment. For When enter 0 and that will send you notifications for any payment made greater than $0. And then on the checkboxes select Invoice. Now you will receive an email and/or text whenever a payment is made on an invoice. 

I hope this helps!