If my AirPods won’t connect to my phone can I get a replacement for free if I go to the Apple store

Great question! So what typically happens is the AirPods will be fully inspected by a tech. Once they diagnose the issue they will be able to replace them if they are still covered under warrant and does not have any accidental damage or been in contact with liquid.

You can verify warrant here: https://checkcoverage.apple.com/

Also the tech will attempt to pair the device so bring in whatever device you are having issues with pairing to the AirPods. Make sure to have a valid ID, proof of purchase, and any parts that came with the AirPods to the service: https://support.apple.com/airpods/repair/service

Only if there is a problem with them.

Only if you are in the Warranty period of 1 year.

Only if you bought AppleCare+ which extends the warranty to 2 years.