I need to get the Messages app on OS X Lion (10.7)
Your question seems to jump around quite a bit and because of that is a little confusing. Below are some answers that may help.


If you want to confirm whether your Mac Mini is 64 bit, the answer is yes.
Your question title says I need to get the Messages app on OS X Lion (10.7) but your question details say you already have the Messages beta app installed. That is the only official version (albeit in beta) that was released for Mac OS X Lion. In other words you already have Messages installed, so I'm not sure what you're question title is about?
Your question also states I need this Mac to run iMovie 11. iMovie '11 was bundled with all new Macs sold on or after October 20, 2010 or for separate purchase in iLife '11. So, if you want it installed on your Mac Mini, you would have had to have purchased and installed it (as part of iLife '11). It was also available for purchase on the Mac App Store. The good news is that in terms of compatibility, iMovie '11 works perfectly in Mac OS X Lion (10.7).


If what you want to know is something different, then please clarify your question.