Can I open my downloaded files in Adobe Illustrator?

Of couse, however, it depends on which Adobe Illustrator version you have and which file you downloaded. Opening a Locator Map SVG delivers a fully vectorised file and all layers can be edited. A PDF is also editable, but there won’t be any layers. The Annotation Map can be downloaded as an EPS which you can definitely open in Adobe Illustrator.
Adobe Illustrator CC and partly CS6 work with SVGs; CS5 and before don’t support SVGs.