Even though *they* say they don't, as the business owner who is delivering the product that people will associate with your business...you should.
The people they give them too might care and if they are delighted with your product, it's more likely that they'll ask where they were done and want to share.
Yes, yes and yes
Even if you found a blank t-shirt for $1 (which may be near impossible to find with cotton prices), with shipping, it would probably be closer to $1.50. Make sure you are not only *covering* your printing and overhead costs, but also figuring in a profit for the job.