Yes, your wild magic continues while unconscious
According to this answer, you can continue to use Tides of Chaos while unconscious. The "unconscious" condition states the following:
Unconscious (Appendix A, p. 292)
- An unconscious creature is incapacitated (see the condition),
- can’t move or speak,
- and is unaware of its surroundings.
- The creature drops whatever it’s holding and falls prone.
- The creature automatically fails Strength and Dexterity saving throws.
Incapacitated (Appendix A p. 291)
- An incapacitated creature can’t take actions or reactions.
Since the wild magic effect does not state any end condition other than "after one minute", it continues and does not itself end because of your lack of consciousness. Additionally, since this is not an action or movement and doesn't require any action on the part of the user, it continues as the Incapacitated and Unconscious conditions do not mechanically affect it.
I would say that from a lore/narrative standpoint this holds up as well. The PHB states on page 103:
However it came to be, this chaotic magic churns within you, waiting for any outlet.
To me, this perfectly fits the magic continuing to function while the user is unconscious, as they're still alive and the magic within them is now free to be released as the user is no longer conscious to control it.
No, magic missile would not work while unconscious.
The entry for the Wild Magic magic missile states:
You cast magic missile as a 5th level spell.
As stated by Szega, you cannot select targets for the spell while unconscious, as you're "unaware of your surroundings". Your 5 senses are not functioning whatsoever (no sight, smell, sounds, touch, or taste).
So as per DM discretion the targets may be randomly selected or the spell may fizzle outright. The PHB states that Wild Magic "creates a random magical effect". I would personally rule that in the specific case of magic missile that it targets random creatures (including the players and the caster) as this is more exciting and in my opinion it is in line with the intended lore and narrative of Wild Magic, but I'm no Adventure League DM.
While I was unable to find any official source for this information, I also couldn't find one that conflicted with what I'm about to say.
This section is my interpretation of the rules. You and your DM may interpret them differently, I would suggest talking with your DM to see what you can work out. The rules for 5e seem to assume that once death occurs, the DM will handle the specific mechanics of it.
I would rule that your character's Wild Magic ceases to function when you die as whatever within you that was powering it (your life force, your soul, some amount of raw magical energy contained within your body) is no longer present.
I would rule that you no longer have a turn when dead, so that Wild Magic (as well as any other effect that does not specifically state that it continues when dead) ceases to function.
Death is death. Once your character dies, it's removed from the initiative order and cannot affect anything else in the game (barring some homebrew classes and mechanics).
As I've said, this is my opinion so feel free to discuss with your DM and come to a different conclusion.
Wild magic continues to function while unconscious but you cannot do anything that requires your character to take any action or use any component to cast the spell. While dead, your character is effectively removed from the game and cannot affect it further. So, to answer your four questions:
- Yes, Wild Magic continues to function while unconscious.
- No, it would not continue if the enemy leader had chosen to kill you outright (subject to DM interpretation).
- You cannot do anything that requires spell components or any sort of action economy while unconscious, nor can you select targets for things such as magic missile. The same applies if you're dead (subject to DM interpretation).
As I said, if any of this information is incorrect, feel free to correct me with a comment and I'll edit the answer appropriately! I hope this was helpful!