Can I request refund for an app that I purchased previously, but has now become free to download? The reason that I would like to request a refund is not just that because it has become free to download.

The app (Splashtop Wired XDisplay), has stopped its support for latest the macOS version, which totally defeats its purpose, which is to allow me to use my iPad as a secondary display for my Mac.
You may or may not be eligible to get a refund. Only the relevant team at Apple can answer that for you.

Consider going through this Apple Support article first: Request a refund for apps or content that you bought from Apple.

You should also consider contacting the customer support for the app to query if they have an update to the app in progress which will enable it to work on the recent/unsupported versions of macOS.

On the other hand, their app is still supported on older (prior to recent) releases of macOS and either some technical or business reason(s) may be preventing them from updating it to work with latest macOS.