No. Your 'Tax Home' determines where you deduct expenses to/from. If you regularly derive income from one location, that's usually considered your Tax Home, and travel to/from there is considered Commuting; Commuting is Not deductible. Where you choose to live has  little impact. For more on the Tax Home concept, see https://www.irs.gov/pub/irs-pdf/p463.pdf 

No, if that one client is your only source of self employment income.  As the other reply indicates, the client's office is your principal place of business and your tax home.

However, if you are regularly self employed, in your home area (you've done this type of work in the past, and expect to do it in the future) AND the current out-of town assignment is expected to last less than one year (and actually does last less than 1 year), you may deduct the out of town expenses and mileage.