Well to add to #1 -- I'd tell you to research and reconsider. No matter how nice your TV the old RCA cables truly murder the quality of what that digital jack on your Cube can do. I was not a believer until I really watched some hi def videos and picture captures enough to attempt a try. Once I had it though damn it was like using the game on the Windows/Android Dolphin emulator. An example the best i can give you, Star Wars Rogue Squadron 2. Look into it. I will give a couple examples. The opening sequence of the game shows a squadron of x-wings and other heading across the screen with one getting very close. The detail on SD is fine, you can make out much of the higher level detail and it's nice and smooth, a little RCA fuzzy but really still a high point. With the adapter though, the detail gets insane. You can see the patterns of the rivets on panels, panel seams, scorch marks tapered off on the metal, the spray of the red paint along the side of the x-wing and so on down to the pilot and stuff in the cockpit. Another would be the famous Hoth sequence with the speeder, I'll just focus on that craft. On SD TVs much like with black/white checkered(small) coats you get this wavy line or rainbow spazz out stuff as it can't properly decipher it as a limitation, so you can see it, it's weird and you can kind of see how it reacts to moving. On the HD adapter you can see every crystal clear independent fan blade and the way they pivot and move as you tip the snowspeeder around. The list I could go on, but the detail makes the GC seem like a lower polygonal WiiU level system, but in games that are more artistic (Zelda, Dragons Lair 3D), cartoony (Mario, other mascots), or even tactically lower polygon due to the space (like star wars) it isn't noticed so the depth is nuts. For $75 it's worth the experiment.
#2 Good move, and yeah clean it well, also get a bit of canned air for the fan blades/housing as it's nastiest and harder to get into the clean. The board and the rest too really since it's so layered and combat due to the cube shape of it.
#3 Tales of Symphonia is a must, also FF Crystal Chronicles would be highly advisable too. Skies of Arcadia Legends as well though it can get a bit more pricey but so worthy for JRPG too. Capcom has their (to the era) entire Resident Evil library on there from the past and that period (Zero and 4) in their finest form of the time, and to perhaps some still outside of HD. Capcom also had SNK vs Capcom 2 which is an amazing traditional fighter. You won't find a whole lot from Square, there are some, just wiki it and see how you feel as it was a period they kind of were aloof about Nintendo consoles while hammering out repeated GBA titles. I would note despite your backlog, you'd be foolish not to get your not dying to play games other than maybe Mario Kart as the ED/FZ/SWRS2 games are undeniably must haves.
When you get if you get around to #4, try the wikipedia, gc game list, sort by JP to the top and see what is only in their region and/or in the PAL area you'll find some really amazing stuff, some that's just inexcusable for not having a US release like Doshin, NIntendo Puzzle Collection, Gundam vs Z-Gundam, Castle of Shikigami 2, and some of those Hudson Select titles (there are 4, I've got Star Soldier and Bonk, Lode Runner kind of sucks sold that and Adventure Island I don't have yet and it's solid.)