What shipping company are you using? I am assuming that your using USPS, and your using the shipping calculator. If so, the resaon that your not able to select First-Class shipping from the drop down is because of USPS, not because of the store. If you want to test if the shipping calculator is working for USPS or not, create an item in the store that is under 13 ounces and then see if the First-Class shipping is an option. USPS only does first class shipping for things under 13 ounces (ie letters, bills, etc.)
This is from the USPS site:
First-Class Mail is required for personal correspondence, handwritten or typewritten material, and bills or statements of account. It may also be used for any mailable item, including advertisements and lightweight merchandise.
For heavier items, see Priority Mail in the Rates and Fees Reference.
Thanks,
Christian