Filing contempt of court motion if other person refuses to be served.?

In most jurisdictions there is a thing called substituted service. You go to a judge, give evidence of attempts you have made to serve and proposed ways to bring process to the attention of the person to be served. The court will make an order that personal service is not necessary and tell you what you have to do to serve. 

You should also check the court rules about modes of service. In my jurisdiction telling a person what the document is and putting it down in their presence is good service. The document does not have to be in the persons hands.

Yes, you can file contempt. Also, he can be served by publication in the newspaper and placed under arrest for contempt by sheriff's deputies.

Can you serve by publication? Or hire a professional process server, many of them are pretty good.

you can try and do that