Did we violate the trademark of the T-shirts by making them into backpacks?

If you do not own the trademark for use on backpacks you may be infringing the use of the trademark on backpacks. A trademark search would be helpful to know if the trademark is registered for goods such as backpacks.

Generally, you should not have a problem. However if you put some scandalous material on say a Jantzen backpack, then the Jantzen company may have an action against you for garnishment of their mark. That is because the person buying the backpack sees the Jantzen trademark on the modified backpack and thinks that the Jantzen company is sell the backpack as it is. The consumer who comes into the store to by the backpack may confuse the source of the backpack.

If the original shirts were "authorized" by the trademark owner - there is less likelihood of a problem, given that your label clarifies the nature of the product as "re-purposed memorabilia." Making that distinction should be enough - but the original trademark owner might disagree. Case law varies based on the specific facts of the matter. The key factor is to avoid any customer confusion as to the source, sponsorship, affiliation, etc. of the goods. Including on each backpack a "created by" tag with your company name on it could be helpful.

Images on T-shirts are generally not trademarks but they may be copyrighted. However, as the owner of the shirts, you are free to do with them as you like, including cutting them up and using them on other items.

Possibly. That one is tricky.