No, our homestay teachers are in Europe on tourist visas so working in the EU is not permitted on a tourist visa. We recommend taking this time to get to know your host family, city, country, and explore!
If you decide to teach to a host family, It is not necessary to have prior English experience. Most of the families want someone who can practice listening and speaking. That could be done spontaneously while performing daily tasks and doesn't require to follow a plan. However, If you are planning to teach in a school, I will recommend having some teaching experience or at least some knowledge...
No, no Italian language experience is required to participate! It certainly helps if you learn the basics before you go though :)