It is impossible to say when the people in Skåne began speaking Swedish. However, I suspect it took a generation or two.
Regarding dialects: All provinces in Sweden have dialects in one form or another. Some dialects are more distinct, others less. Some distinct dialects are the Värmland dialect, Dalarna, Gotland, and all the various dialects in Norrland, to mention a few.
And yes, the Skåne dialect is quite distinct.
However, I think dialects are less and less spoken nowadays, which is sad. Young people tend to speak National Swedish, what we call “Rikssvenska”.
The Scandinavian languages are quite similar, Swedish, Norwegian, and Danish. We understand each other, although Norwegian is easier for Swedes to understand than Danish, but it is possible.
/Hans