Hi there, I haven't read any of the other posts, but I'll try and answer your questions, please let me know if I miss one. To be up front, I've worked with computers for more than 10 years, I've been a PC 'user' for over 15, and have built 3 new computers and worked on several in the last 5+ years. So I may have some small experiences relevant to your queries.
In reverse order, yes, connecting your 47 inch television to your computer should be quite easy, you'll simply need to have a graphics card with an 'hmdi' out port, or an adapter which depending on your budget may be cheaper to skimp on the GC and get the adapter afterward, DVI and HDMI are the same quality signal, HDMI simply includes audio as well (forgive me if this is obvious).
For the Power Supply, you do have a nicely efficient system in place there, and a 'single' graphics card shouldn't require too much power, I'm guessing you currently have a 400W PSU, perhaps slightly higher, I'd recommend (for future proofing reasons) to buy a 750 PSU+ you may be able to get away with a 650 PSU.
The big question is the graphics card, now I've been a loyal Nvidia user for many years, my current card was the best single graphics card of the DX 10 era, the 285GTX, but I would not recommend Nvidia for this era (DX 11) as their technology per price is neither superior at the top end, as it usually is in the past, nor better priced in the middle. I think the best card for the price point would be the 6950, its the second best single card they offer, but it can easily be changed to perform almost as well as the 6970 (depending on brand, you can look up which ones you can normal clock).
As you identified, you will need a bigger case, a mid-tower will be gigantic for your current MB, but should also be perfect for any future normal size ATX MBs should you wish to recycle that case in the future (it'll also hold the new PSU nicely).
I cannot recommend the RAM too well, make sure its as fast as the CPU or faster, I haven't researched any of the i3s, but a 3.07GHz i3 must be one of the better ones, I'm going to guess its front side bus (the speed in which the processor communicates with the ram) would be atleast 1066MHz, perhaps 1333MHz, so get ram that maximizes that, confirm whether or not your MB can handle the amount you want, 6-8GB should be a good amount, 2 or 4GB modules (if the MB supports), are a good idea, preferably 4GB as they'll last longer.
Any budget? eBay is a good bet for parts, but be careful, I've bought a MB from there before and it didn't work because it was used and mistreated.
Let us know how you go, hope I've helped and answered all of your questions.
