To answer your first question i have looked at the page we have on
Denver Hotels. This page mentions a few hotels in the area but i wouldnt say they sound like 'budget' hotels. However i know a cheaper option would be the Best Western Denver Southwest or the Country Inn and Suites. These arent situated in the centre of Denver but perhaps it might be an idea to look at hotels a little further out of the city as this is where you will find the cheaper option.
When in Denver i would recommend visiting the
Museum of Nature & Science and the
Denver Zoo.