I would like to have a "New" products carousel on my home page. I've got them for other items in specific categories but I would like one that lists all items that are new regardless of category.
No Magento Will not Provide you the Default Category ID for new category product.
1) you can do one thing create the new category like
2) When you create new products then assign that product to that category as well as your normal category.
3) At Home page just load the category and make slider as per your requirement.