Welcome, Marco!

I don't know about gig, but someone here will..or will research it for you.

For woodshedding, I can take an educated guess. When in the old days farmers went out to the woodshed, it was to chop felled, mostly seasoned trees into logs for the stove. It was hard work, it had to be done, and it was necessary to sustain life.

For a musician, woodshedding is getting the work of practice done, the working out of parts, difficult passages, desired effect, and so on, to lead to eventual performance. Woodshedding was analagous to the work in the woodshed for the farmer, difficult and demanding, but, oh, the resultant fire!