I am an independent contractor by definition. I work out of an office in which is over 80 miles one way from my home to the office. Can I claim gas used to commute?

If you go to an employer's office to work you are an employee and the commuting expenses are not deductible.

There is where the confusion lies. I filed a W-9 when this employer hired me, which labels me as an independent contractor. Can I still not claim anything at all?

No. Commuting is not deductible, even for the self employed (the W-9 is not relevant).  But, the factor of "principal place of business" enters the equation. A client's office can be your principal place of business; in which case your mileage is commuting and not deductible. To deduct mileage, your home just needs to your  "Principal Place of Business". See reference links below.

If you go to 2 or more job sites in a single day then the 2nd trip is deductible as an employee expense. For example you leave home drive 80 miles to the client's office then drive 10 miles to the 2nd site. At the end of the day you drive 85 miles home. Only the 10 miles between jobs is deductible. The first trip in the morning, from home and the last trip at night back home are considered commuting and are not deductible.

  
https://www.irs.gov/pub/irs-drop/rr-99-7.pdf
https://www.irs.gov/publications/p463/ch04.html

So as far as calculating income and self-employment taxes, I understand that I pay 15.3% under self-employment taxes, but how much additional do I pay for income taxes?

Your income taxes are calculated the same as they would be if you were a W-2 employee. They are based on your adjusted gross income less your standard or itemized deductions and personal or dependency exemptions. That is your taxable income which the tax is calculated on.

I have read elsewhere that if the work is temporary (less than 12 months) then mileage is deductible.

Maybe, if under the right circumstances.
First, it was provably an old answer, for tax years prior to 2018. W-2 employees are no longer allowed to deduct the cost of out-of-town trips. Even in the old days, certain conditions had to be met.

For a self employed person to deduct the mileage, he would have to also be self employed  in his home area. That is, he could not deduct the mileage for a one time gig, even if temporary

Traveling from home to your main job is commuting and non deductible regardless of whether you are self employed or an employee.