If I do my illustrations on 11x14 paper, can Lulu reduce them to fit an industry standard children's book size?
Ron has already answered that, but perhaps to expand on it, you need to look at the size options available in Create. Lulu is mostly a DIY self-publishing site and Create is the start of uploading pre-created files of pages to a Project Wizard to match all the specs in Create.
Can a do illustrations as a two-page spread?
You can if you understand how to create Full-bleed pages in something such as Word, using the sizing advice in Lulu's Knowledge base.
Or do illustrations have to be done to fit single pages?
Sort of yes, and no. To do a double page spread an image has to be cut in half, and each half placed opposite each other on two pages. So really you are still dealing with images on single pages.