Did you have tires fully inflated? Sound like that could be part of the problem, given you don't have a pump. I have to add air (top off) to my bike tires just about every day.

Go to the clydedale forum, and you can find some awesome advice on wheel choice.

+1 Find another shop, depending where you live, there should be several decent choices. And the shop doesn't have to be that close to where you live or that convenient, as you probably won't be going there that much anyway. My favorite shop is 75 miles one way from my home.

What is your goal with the tri bike? Depending on how serious you are about it, a decent road bike might meet your needs just fine.

I would like to see a picture of your valve stem, as you may just want to change out the tubes? Are your tires tubulars or clinchers? Most decent bikes come with presta tubes. If thats what you have, thats not really a funky tube, thats normal. Get a bicycle floor pump, you will need one regardless.