According to the BBC, there was 279 councils in England in 2015, versus 124 in 2016. Why is that number different?
As mentioned in this question, councils in the UK can be elected all at once, or have half or a third of their seats up for election at a time, with 1, 2 or 3 elections in every 4 year cycle, respectively.
As a result, the number of councils with seats up for election in 2016 is different from 2015.
the UK in general and England specifically is currently predominantly Conservative
Well, not really; they received just over a third of the national vote in 2015.
Why is it the case then that most councils in England are won by Labour?
Because the spread is not uniform. Very broadly, Labour tend to do better (in England, at least) in urban areas, while the Conservatives tend to do better in rural areas. This map illustrates that quite nicely.