For question #1: A traditional 401K reduces your income, but not your FICA taxes, which means the employer must first deduct the employee portion of FICA from your paycheck, and the remainder is then eligible to be contributed to your 401K. Interesting idea but I don't believe you could offer to pay the FICA taxes on the side to increase the contribution amount.
For question #2: Yes, the employer contribution can still be made, and since it's typically based on the employee's gross amount, it would not be affected by how much the employee contributes, as long as the minimum threshholds are met.
As a side note, if this is the employee's only income, they may be better off contributing to a Roth 401K, as the income tax amount would be low, possibly $0 after deductions.