Question 1
Its well known that if A Paladin commits a crime that even if he or she isn't convicted he or she still loses his or her powers but what happens if A Paladins is convicted of a crime that he or she didn't commit?
Well, no. In 5e, it isn't a question of whether the paladin breaks the law. It is whether they violate their Oath.
For my games, it has always been such - the laws of mortals can be corrupt. The paladin must do the right thing (as defined by their Oath) whatever the law may say.
Question 2
If being convicted of a crime that he or she didn't commit doesn't cost A Paladin his or her powers would demonstrating that he or she still has his or her powers get the conviction over turned?
Conviction by a mortal court is not material to a paladin's standing. Courts can be corrupt, or incorrect, or just irrelevant to the paladin's moral standing.
Also, as above, the Oath is the operative thing - the paladin may be guilty of a crime, but not guilty of violating their oath. If, for example, in a war a paladin discovered that the war, battles, and leaders were unjust and wicked, so the paladin leaves the fight, the might be guilty of desertion in the legal sense, but entirely in good standing with their Oath.
Question 3
If A Paladins God or Goddess not only gives him or her A Quest but also tells him or her that if during this quest he or she needs to violate the paladins code then he or she can do so and won't lose his or her powers would this allow him or her to violate the paladins code without losing his or her powers or wouldn't it?, after all even gods and goddess make mistakes and if this happened then this could be 1 of those times?
First, the gods almost never speak directly to mortals in my worlds. Paladins and other people with divine powers exist
because the gods cannot or will not intervene directly like that.
Second, especially in 5e, paladins don't actually have to be tied to a god. It is their devotion to their Oath that allows them access to the power. The paladin violating the Oath will lose powers, whatever the god may say or want.
Question 4
If A Paladin worships a very tolerant god or goddess that will allow a paladin to commit a fairly large variety of the ends justify the means acts without losing his or her powers would that include things such as saving 10 of thousands of lives by killing a few hundred people that have the plague?
See above that the deity isn't actually the relevant thing.
Most paladin Oaths won't allow such as the above.. I can imagine an Oath constructed to allow such, but it will have just a strong strictures in some other way.
Question 5
Since Paladins that commit certain acts lose the powers of a Paladin but in some situations f they accomplish a sufficient act of redemption they can regain the abilities of a Paladin does this mean that if a sufficiently devout Fight commits enough notable deeds that he or she can earn elevation to the status of A Paladin?
Good deeds alone do not make the paladin. It is, again
devotion to the Oath that makes the paladin.
Question 6
If a very young Paladin saves A Toddler and many years later that Toddler becomes 1 of the worlds most powerful servants of Chaos would The Paladin lose their powers even though they had no way of knowing that The Toddler they saved would grow up to become 1 of the worlds most powerful servants of Chaos?
A paladin is not responsible for the choices of others made well after their interaction.