Your employer is not required to pay or reimburse any cost of being an employee. This is a matter of your employment contract and is something you should negotiate.
If the employer decides to reimburse using a non-accountable plan, the reimbursement is part of your W-2 box 1 wages. If they reimburse under an accountable plan and they reimburse at MORE than the IRS mileage rate, the difference is taxable income to you. If they reimburse under an accountable plan and pay less than the IRS rate, the difference used to be a miscellaneous itemized deduction subject to the 2% rule, but that deduction was eliminated by tax reform.
As to the question of whether your time while traveling should be "on the clock" (that would be wages, and not a mileage reimbursement question) that would be a matter for state law. You may want to consult a labor attorney or your state labor commission.
Regarding costs of being an employee, that may also be a matter for state law. For example, if the employer requires you to wear a certain uniform, they may be required to pay (or reimburse) for it. But there is nothing in the tax code that requires employer reimburse anything. The tax code only describes the tax treatment of different kinds of reimbursement.
