Is there a way to change the state of a dropdown in a form via a link click in an email?

Yes, but you will need a formula that users a Usertoken and App token to bypass permissions that normally require a user to login.  Build it as a formula URL field to make the API_EditRecord.  Include the usertoken= and apptoken=.

If there are multiple "choices" to select; then you will need a set of formula-fields; or if you know how you can create a rich-text formula that strings together separate buttons in a single field for the same purpose.

I recommend a robot-user for the Usertoken; you do not want a button to break because your Usertoken belongs to a User who has been denied from your account if they leave your business.

Naveed, if you do want to Authenticate with a robot userid, post back and I can help you with the code.  But note that you will not be bale to know who really clicked the button if you don't require the user to log in.