Income tax for payments as a part of job application tasks

So I should fill out Schedule C with my profession as Unclassified establishments (unable to classify)?

I'm assuming you're applying for jobs in one specific field, and the odd tasks you've been asked to do are related to that field? So use that.

Another option is to include this on Schedule 1, item 8j, as activity not engaged in for profit income.

No, that would be wrong since you were in fact engaged for profit income (you're looking for a job, not doing charity work or hobby work).

Another option is to include this on Schedule 1, item 8z

That's somewhat reasonable. You would still need to include this under self-employment in Schedule SE and pay the applicable taxes, so Schedule C might be better to automate that part in your tax software.