Can anyone recommend a free extension that allows a carousel to be placed on the hompage with rotating images?
The images should be able to be uploaded in the admin area.
I would recommend Qbanner by Qaz. It works very well and is easy to customize.
A big plus is the admin section that offers a lot of options for placing the banner without the need to edit layout files and it offers a nice grid for the uploaded slides.
We did replace the frontend javascript with jQuery Cycle slider to make it responsive. This is also easy to do by copying app/design/frontend/default/default/qaz/qbanner/qbanner.content.top.phtml to your local template folder and replacing it with the code found on the cycle slider website using the Qbanner functions to build the HTML.
There are several available at Magento Connect. I did a search for the word "carousel" and chose "free".