Anyone who meets the criteria and requirements set by the European countries can work in Europe. Most European states have their programs, through which they tend to lure foreign professionals and fill job shortages in different fields in the country.
The Schengen work visa does not exist. You can get a Schengen visa for other purposes, as tourism, visiting family and friends, business, medical purposes etc. However, you cannot get a Schengen visa to work in Europe. You are not permitted to work while holding a Schengen visa for other purposes, as well.
Still, you can work in the Schengen Area if you hold a National (D) Visa for employment purposes issued by one of the 27 European countries part of the Schengen Zone.
Each of the Schengen member countries has its own visa policies, which policies differ from one country to the other. The employment visa programs in the European countries have been established to cover the labor needs of the respective countries and fill job shortages.
Therefore, employment visa criteria and requirements, as well as the application process, depend a lot on the labor needs of each country.
Citizens of the USA, Australia, Canada, Israel, Japan, New Zealand, Switzerland, as well as EU citizens do not need to apply for a work visa to Europe. However, upon arriving at the country where they will be working, they have to apply for their residence and work permit.
Citizens of other countries must apply and get an employment visa before entering the Schengen territories for work purposes.
You will be able to extend an EU work permit in most countries. There is an application process and some required documents, which you will have to collect and submit at the competent authorities in your European country of residence, prior to your visa / residence permit expiration.