I just setup a small business and its barely a week old. I have applied for sales tax number. Now, i am employed in a company, so this business is in my name.
Business is in wife's name but she already has a beauty parlour under her belt. Does this affect the current application for sales tax?

Nope it shouldn't affect
Nope it shouldn't affect anything. As long as the business numbers are different, the naming should not be a problem at all.