I'm not sure what a sticky post has to do with it. Simply put, a sticky post is a post that will stay at or near the top of the last published posts. So if you have a "last post" page that you use as the frontpage, your stickied posts will be there.
Now, if you want tabs somehow in there, I'd suggest to put them in a section of the template that is not touched by the posts, stickied or not. That's what you can arrange within the theme; edit or create a "homepage" page template (and CSS styling) to do what you want it to do, where you want it.
The alternative is to change the architecture: use a "regular" front page and use it to show featured posts in another way, there are tons of featured post plugins out there. Then you can put the tabs wherever you want on that page. Here's an example: http://www.nitaonline.org
It uses GT Tabs as well as Featured Articles Lite (a featured posts slider plugin) on a "normal" page set as the frontpage. Note: it's not the aesthetics or design but the functionality, which is why I point to that place.