A.  What is the connection between Senator Palpatine and Darth Sidious?

According to Cinescape Online's Insider, Rick McCallum, TPM's producer set the record straight on Sidious' identity (also check Question 3I) in a British magazine called Starburst by saying "Ian [McDiarmid] is the Phantom Menace. He's not the Emperor yet [in The Phantom Menace], but he is a Sith Lord. He's also Darth Sidious. That's Ian in the robe."

George Lucas himself has stated in a USA Today article that the outline for Episode II includes the story of Darth Sidious rising to power in the senate and becoming emperor. We all know that Senator Palpatine and the Emperor are one and the same from before and know it is he that rises in power in the senate.

In addition, it dosn't hurt that they are both played by Ian McDiarmid, who played the Emperor in ROTJ (didn't you get a chill down your back when the good senator called Anakin "Young Skywalker." while putting a hand on his  shoulder?)
 

B.  Can you explain the whole Queen Amidala/Padmé thing again? Did they have two different accents?

For protection, Queen Amidala assumed the identity of Padmé (which is the Queen's real first name), the Queen's handmaiden. The girl posing as the queen was, as described in the movie, a decoy and bodyguard. 

It is unknown if the Queen spent the whole length of the movie posing as Padmé. She most probably did as a precaution at the beginning due to the imminent invasion of Naboo.

As for the accents, Padmé (the true queen) had the American accent while the decoy dressed as Queen Amidala talked in a light English accent. Natalie Portman, the actress that played the Queen/Padmé, has an American accent.

The decoy also seems to be played by another actress. Named Sabé in the novelization, the decoy is not named during the movie, but the name Sabé appears in the credits, to that of actress Kiera Knightley (along with Rabé, Eirtaé, Yané, and Saché, Amidala's other handmaidens.)

(The Queen Amidala FAQ bases itself on many different Episode I books and products as much the movie itself and has an explanation of when the Queen is really the queen, and when the decoy is impersonating her throughout The Phantom Menace, based on the novelization. Also, so far, story-wise, Queen Amidala and her decoy do not seem to be clones of each other.) The FAQ also seems to focus a lot on the clothes Natalie Portman wears in the omvie.
 

C.  Why didn't Qui-Gon Jinn's body dissapear upon his death, like Obi-Wan, Yoda, or Anakin?

It's a good question considering the importance of Obi-Wan's dissapearance. While some people have attempted to guess, they are once again speculating.

George Lucas has said in an interview that there is a reason for this, and it will be revealed in Episode II. Lucas also said that there is a clue in ANH when Obi-Wan tells Darth Vader "If you strike me down, I shall become more powerful than you can possibly imagine."
 

D.  Could you explain Senator Palpatine's rise to power again?

Someone asked this one, folks...

It is not known whether Senator Palpatine/Darth Sidious foresaw that the invasion of Naboo would result in Queen Amidala's escape to Coruscant, where she would make a plea of assistance to the senate.

Whether a coincidence or not, Palpatine used this as an opportunity to convince the Queen to motion for a vote of "no confidence" in Chancellor Vallorum, head of the senate. (A vote of no confidence is a motion to elect a new senator as Chancellor.) Palpatine was nominated a candidate and then won the election. (Partly due to "sympathy" from people due to the Naboo situation.)

Though under Palpatine's control from this point on, the Senate continues to exist in some way or other until he completely dissolves what is by the end referred to as the "Imperial Senate" in Episode IV: A New Hope. The rest of this dastardly plot is revealed in Episodes II and III.
 

E.  If there are only two Sith at any given time, what does that mean in terms of the Original Trilogy?

"Only two there are. A master and an apprentice." - Yoda

The question is raised because if there can only be two Sith, what would have happened had Luke Skywalker been turned to the dark side? Obviously, either Darth Vader or the Emperor would have had to have been eliminated.

In ESB, Darth Vader tempts Luke by offering him the opportunity to overthrow the Emperor and "rule the galaxy as father and son." In ROTJ, the Emperor tries to lure Luke to dark side of the force and to take his father's place. It's clear that both Sith would be prepared to destroy the other for a chance at gaining a more powerful apprentice.
 

F.  Who is Anakin's father?

Luke had a much easier answer to his question by comparison.

Shmi Skywalker claims that Anakin has no father. She bore the child, gave birth to him, and raised him. (In case some of you don't know, this sounds eerily like the Virgin Birth, the Christian belief that Jesus Christ had no human father, but conceived solely by the Holy Spirit.)

Qui-Gon Jinn claims is that he is the Chosen One, conceived by the Midichlorians, which brings us to our next question...
 

G.  What are Midichlorians?

Midichlorians are microscopic life forms that exist in all living cells. Without them life could not exist, and living beings would have no way of tapping into the Force. (How they got into "the rock... the ship..." is beyond me...)

Luckily, they were not mentioned in Episode II.
 

H.  What the hell are the words to "Duel of the Fates?" What language is that?

John Williams chose Sanskrit as the language in which the choir would sing the song. It is an ancient language originating in India, and as he states, is less known than Latin or Ancient Greek.

Worminator was very helpful and had this to say:

The lyrics to Duel of the Fates are translated from a famous Celtic/Druidic poem whose title is roughly "Battle of the Trees" The lyrics are actually one line of this poem repeated over and over which has something to do with two battles taking place at the same time.

Since this e-mail I've found a copy of "Battle of the Trees" online at http://dandalf.com/dandalf/cadgoddeu.html . It seems to be a bit longer than the translation to Sanskrit. Maybe they just took a verse. The poem is a story from what seems to be Arthurian legend.

I.  What is The Phantom Menace?

The Phantom Menace refers specifically to Darth Sidious, but the movie does suggests that it's the mysterious resurgence of the Sith in general. So far, all the titles except ESB (and now "Attack of the Clones") are a reference to matters of the Jedi and the Force, even if theirs wasn't the principal story of that particular movie.
 

J.  Does Anakin/Darth Vader really bring balance to the Force?

Quite the philosophical debate. Some people think that Qui-Gon COMPLETELY missed the target. George Lucas has answered this question in an interview. He claims that it is Anakin that restores balance, for it was Darth Vader, and not Luke, who destroys the Emperor. Supposedly, only he had the power to accomplish this task.

This brings up a wonderful question as to whether Yoda and Obi-Wan's original plan was for Luke (or Leia) to turn Vader against his Emperor, as neither was powerful enough to stop him. Obi-Wan states in ROTJ that Vader cannot be turned, which would make his and Yoda's plan to destroy the Vader and the Emperor with Luke a flawed plan, seeing how things worked out.

The Republic was a bureaucracy, and the Jedi Order did have its flaws. Corruption in Republic politics is what first allowed Darth Sidious to take power, and the pain of putting emotions aside, a Jedi practice, is what probably led Anakin Skywalker to the dark side. The Emperor and Darth Vader wiped the slate clean, and then they themselves were wiped out. How much more balanced can things be?