1) What are your most recommended games for the system that aren't available on Wii, Wii U or 3DS Virtual Console? And why are they your faves?
There are too many good games on the Genesis. Here's an extremely long list of games I recommend.
Shining ___
Sonic ___
Aladdin (Better than the SNES version)
Castlevania Bloodlines
Castle of Illusion
Contra Hard Corps
Earthworm Jim (better than the SNES version)
Ecco the Dolphin
Golden Axe 1, 2, and 3
Ghouls n Ghosts
Gunstar Heroes
Landstalker
The Lion King
Maximum Carnage
Monster World 3 & 4
Phantasy Star 2,
4 (start with 4 then do 1-4)
Ristar
Shinobi III and Revenge of Shinobi
Sparkster
Street Fighter 2 Championship Edition (better than SNES)
Streets of Rage (especially 4)
Thunderforce
TMNT: The Hyperstone Heist
Toejam and Earl
Truxton
Zombies Ate My Neighbors (not as good as SNES)
2) What is the best way to play the system on an HD TV?
If a PVM is not an option, get an XRGB-3, a Framemeister or an OSSC.
BTW If you are using a Genesis 3 (that tiny piece of shit in your picture) ignore the previous sentence. You are going to have to deal with a bad experience (sound, video, ability to handle expansions). When you get a chance, get a model 1.
3) If we want the ability to save games, is a GameShark our best bet or will most games have a level select code? Forgive my ignorance, but do any games offer a save feature?
Don't do this. It ruins the games. Accept your loss and keep trying.
4) What's the region locking situation? Can I play JP games on the system?
Japanese Genesis games are a slightly different shape. You can play almost all of them by making a slight change to the shape of your cartridge slot or using a Game Genie. Games that have further region locking can be bypassed by a simple mod or using a Game Genie code. PAL games require a hardware mod.
5) Are some of the Disney games that are also on SNES, i.e. Lion King, Aladdin, essentially identical to their SNES counterparts or are completely different games?
All Disney games are better on the Genesis.