I currently use the Evo 4G LTE (have the Evo 3D in a drawer - not going to update), and to turn off background data you HAVE to set a data limit and then go into settings (on screen menu key since there is not a hardware menu button anymore - for you try hitting the menu button) and select turn off background data. I believe this is an Ice Cream Sandwich change and should be similar across the handsets even though Sense is a different version?
If successful, you will probably have a notification in your bar all the time showing that background data is off.
As far as getting the phone to open applications instead of websites, you may want to give this thread on the Evo 4G LTE forum a read through:
Android Forums | Evo 4G LTE - How to get it to open apps instead of websites?
Always on data option is gone as far as I can guess, based on my similar phone. The closest thing I can think of now is in the advanced wifi settings you can specify to keep wifi ono during phone sleep, and use best performance which consumes more power.