Is it permissible to deal with a hired worker who refuses to specify the wage before the work, while insisting on this from the worker's side? Will I incur sin by doing so? And is it permissible to agree upon the wage while giving a portion of it before the completion of the work?

There is no harm in a person engaging someone who presents himself for work as a tailor or similar without a contract. The customary wage is due upon completion of the work. It is permissible to pay the wage or part of it in advance, for the important thing is that the worker receives his full wage, and it is not a condition that the wage be paid after the completion of the work.