Although these questions can usually generate the next great OSR controversy in certain corners of the internet, here is my take on them.
Race (Elf, Dwarf, Halfling) as a class?
No. This is mostly because there are a lot of options for race and class in anything I run. There are the typical choices for race, Dwarf, Elf, Halfling, and then a few others that make sense in the campaign world. For all these races to survive, each group has a specialty, but individual members of a particular race can cover all kinds of rolls.
For example, Elves specialize in magic and this makes them powerful mages and more resistant to magic than most races. Their extremely long lifespan makes them expert craftsman of both magical and non-magical objects. However, not all Elves are magic-users. Some are Clerics to be an intermediary between the Elves and their gods. Some are Thieves, though I do Thieves differently, so it isn't about Pick Pockets and Backstabbing as much as other skills such as Appraisal, Climbing Surfaces and other things. Some are Alchemists while others are a different kind of magic wielding class with a very different method of employing spells.
Do demi-humans have souls?
Yes and no. Elves, dwarves, and halflings have souls by default. One race in my world is called the soulless. They have traded their soul for an immortal body. Created beings, with one exception, do not have souls. This includes golems and undead. Regardless of intelligence, they do not have souls.
When anyone dies, they go to the same afterlife, its just that different areas of the afterlife look different than other areas. It's just like the Prime Material Plane in a sense: there are humans in one part of the world, Elves in another, halflings here and there, etc. Races don't separate to go into their own racial afterlife, I use alignment as more of a determinant.
Ascending or descending armor class?
Demi-human level limits?
No. Some people are better at things than others by racial trait, superior ability (like INT 18) or skill (like a dwarf's stone-cunning abilities).
Should thief be a class?
Yes, but with more options. A character chooses which five or six skills they want and play from there. This is similar to how the 2e PHB does Thieves.
Do characters get non-weapon skills?
Thieves have them and so do my other classes. My Fighter class, for example, has the ability to catch/deflect any thrown object. With so many class options, the non-weapon skills are built into each class.
Anyone can try to do anything and have a chance of success. Want your Magic-User to pick a lock? The odds are less than a Thief, but go ahead. Want to be a baker? That's fine. Rolls probably won't be needed unless you want to do something extraordinary like make a self-slicing cake. (My favorite is the cake that is larger on the inside than the outside - it is essentially a Cake of Holding where all the storage space is filled with cake.)
Are magic-users more powerful than fighters (and, if yes, what level do they take the lead)?
Yes. Magic-Users, after level 7 or 8 become a force to be reckoned with. They are impressive and not to be trifled with.
Do you use alignment languages?
I would love to, but I never understood it well enough to get it to work. I just can't find an in-universe reason for these to exist.
XP for gold, or XP for objectives (thieves disarming traps, etc...)?
Yes. XP for finding and spending gold (half for finding it, half for spending it). I also have XP for finding certain areas of the world, or certain mythical objects (both magical and non-magical). I do have XP for a few objectives, but that depends on the session. I don't want to be railroad-y, so anything related to XP for objectives tends to be more like finding certain things.
Which is the best edition; ODD, Holmes, Moldvay, Mentzer, Rules Cyclopedia, 1E ADD, 2E ADD, 3E DD, 4E DD, Next ?
I never played only one edition. When I started playing, I owned B/X, 1E PHB and DMG, the Companion Rules (but not Master), and the Spelljammer Box set. I also had 2e Shadowrun and Baattletech. I don't even know what version to call it except D&D Mine. (To coin a phrase from someone else.) Currently, I like Swords & Wizardry because I can fit all of it into the ruleset, including Battletech.
Bonus Question: Unified XP level tables or individual XP level tables for each class?
Individual. XP is currency in all my classes and races. You want to use a double-bladed axe, hurl fireballs, heal the masses and wrangle ghosts and other incorporeal creatures with your bare hands? That will cost you XP.