Efficiency
No one anywhere can require you to sign anything. Doing so would invalidate your signature. It must be voluntary.
No, the employer does not have to prove anything. The organization can require whatever they feel is appropriate for the position to be filled. They look for the best fit, and thus establish the educational criteria particular to that position.
No.
Although the question is nonsensical in part, an employer can require you to do overtime if that is what the job needs. However, if overtime is not needed, you do not work it; overtime is not a part of the job, so you are not taking anything off.
Yes, they can. It is possible that the training expenses would be tax deductible. You would need to check with a tax professional regarding that aspect of course. Otherwise, payment should be made on a 60-40 or 50-50 basis where in 60% will be paid by your employer and the remaining is to be paid by the employee.
Legally, though local laws may vary, the employer can require the employee to report to work in a uniform without providing any assistance to the employee to acquire that uniform. Ethics of this situation are debatable. Ultimately, either you want to work there or you don't.
Can your employer require you to have your picture taken to download to the company computer system?
YES
who shuld no
yes
No employer can require things being done with or without pay. It is against the labor laws which are given by the federal government.
Usually you would receive specific training for the chain hoist trolley provided by the company you are working for. Some states however may require you to become certified with the equipment beforehand which will also be provided by your employer.