Do law essays count towards your degree?

Yes, law essays (or coursework or assessments, however you want to put it) do count towards your degree. This will be based on what modules you choose for your 2nd and 3rd years too as some will weight more heavily on the essays rather than the exams and vice versa so you have some freedom to choose your best option.

Most of the time, first year also does not officially count towards the rest of your degree unless you are planning to get apprenticeships/internships/work experience as they sometimes look at how hard you worked in your first year.

In short, yes essays count and are assessed and they are part of the degree, and no usually the first year does not count towards the final degree mark.