Can I send events to target pages?

I would like to be able to script sending events like mouse clicks and keypresses to the sites under test, ideally using selectors. e.g. send “myame” to ('input#username'), send "mypass" to (‘input#password’) and click the button that says “Login”.

Is there a way to do that with Postman?

No.

That is UI Automation. You might need to look into something along the lines of Selenium.