I would like to know if there are researchers conducting studies on public opinion on regional integration - whether in the European Union or in other regions.
There is a rich host of studies concerning European integration beginning in the 1950s. The United States Information Agency began asking citizens of the original European Coal and Steel Community in 1952. Based on that experience and some forerunners, the Information Service of the European Commission set up the Eurobarometer survey series in 1973. All surveys are available at GESIS:
http://www.gesis.org/en/eurobarometer/home
The European Social Survey (ESS) asks regular questions concerning European integration:
http://www.europeansocialsurvey.org/
There are literally hundreds of articles on "public support", "attitudes toward", or "public opinion" concerning European integration. Sara Hobolt's chapter in the 'The Oxford Handbook of the European Union' gives you a good overview of the topic: