Yes you can remove categories by deleting them from “Manage” -> “Categories” page. After deleting all categories category field will not be displayed when posting a new ad.
Displaying custom search fields on home page is currently not available we plan to add it in our next release. (planned release in 24 hours)
Listing ads without any category is not available as well, we plan to add it in our next release. (planned release in 24 hours)