The purpose of sweet 16 invitations I think are to invite people to a gathering known as a sweet sixteen event. These events could be social and/or formal or informal.

Same as Sweet Sixteen parties. If you cannot find pre-printed ones where you live, get blank party invitations and fill that in yourself. For your non-Hispanic friends, tell them it's like a Sweet Sixteen party, but in your culture , you celebrate this step to womanhood a year earlier.

If you are choosing to make your own then check out wikihow.com or instructables.com. If you are looking to buy sweet sixteen invitations, the you should try vistaprint.ca. They have all the invitations you need, and you can also design them yourself to add a personal touch.

"des invitations d'anniversaire / des invitations à un anniversaire"

Yes, the noun 'invitations' is a common noun, a word for any invitations of any kind.