Is there any way to add "Send & Delete" button in GMail like "Send & Archive" which can be enabled in labs. any add-ons, extensions or greasemonkey script will do.
There's nothing in Labs to do this at this time.
Depending on what you're after, you might be able to get close using filters. Create a filter that looks for a particular string in the content ("Contains:"). Make the string something innocuous that no one would ever think to put in a message (something like
When you send a message with that string in it the message will get labelled "Trash" (as if you'd deleted it yourself).
Two possible complications
Depending on what you're trying to accomplish these may or may not be a problem.