Well, I'm nowhere near fluent in Japanese, but let's see what I can do.
Please note that in some cases, this is a "gisted" translation--not a direct translation of the question.
Q1 <<¥1,000>>
In the game of shiri-tori, which of the following words would cause you to lose?
A: Sukii (skiing)
B: Yukidaruma (snowman)
C: Mikan (tangerine)
D: Kotatsu
Shiri-tori is a wordplay game where one person says a word, then the next person has to say a word that starts with the last syllable of the previous word. Anything that ends in the -n sound (ん), however, automatically loses, because it's impossible to start a word with that syllable.
Q2 <<¥2,000>>
In the house, where are you most likely to do cooking (ryori)?
A: Genkan (entrance)
B: Toire (toilet)
C: Kitchin (duh)
D: Ofuro (bath)
Q3 <<¥3,000>>
Which of the following animals has a shell?
A: Kaeru (frog)
B: Tokage (lizard)
C: Hebi (snake)
D: Kame (turtle)
Q4 <<¥5,000>>
Which of the following equations adds up to 10?
(choices self-explanatory)
Q5 <<¥10,000>>
In Japan, which prefecture is the northernmost?
A: Hokkaido
B: Chiba
C: Osaka
D: Fukuoka
Q6 <<¥15,000>>
Which color results from mixing yellow and blue?
A: Midori (Green)
B: Murasaki (Purple)
C: Kuro (Black)
D: Pinku (duh)
Q7 <<¥25,000>>
Which comedy duo was famous for "Natural Gymnastics"?
A: Piisu (Peace)
B: Kau-kau
C: Harisenbon
D: Hanniya
I'm not sure what this refers to--a Google search didn't turn up anything, but then I wasn't really looking because I was translating the rest of the questions.
Q8 <<¥50,000>>
(This one's tricky. I think it's something along the lines of "in the word 'nail salon' (ネイルサロン), how do you write "nail" in kanji?" Nail, obviously, in the sense of a fingernail or toenail--you might hear Mana-chan mutter something about a manicure when she's pondering this question. The four kanji you see on screen were read with their kunyomi readings, confounding the matter even further.)
Q9 <<¥75,000>>
What was the name of the princess in Sleeping Beauty?
A: Jasmine
B: Belle
C: Aurora
D: Ariel
(The STUDY graphic points out which story each princess comes from.)
Q10 <<¥100,000>>
Who is the current American President (daito-ryo)?
A: Putin
B: Charles
C: Obama
D: Clinton
Q11 <<¥150,000>>
Which of the following is, simply, fish eggs (sakana no tamago)?
A: Foie gras
B: Caviar
C: Truffle
D: Fukahire (shark's fin)
After the question, Mana-chan is asked whether she's ever had caviar before, to which she replies "no"...but then surprises Mino-san when she knows that foie gras, caviar, and truffle are the three delicacies of the world!
Q12 <<¥250,000>>
Which of the following is the name of an Egyptian queen?
A: Marie Antoinette
B: Mother Theresa
C: Cleopatra
D: Jeanne d'Arc (Joan of Arc)
Q13 <<¥500,000>>
The founder of the Nobel Prizes, Alfred Nobel, invented what?
A: Denkyuu (Lightbulb)
B: Rentogen (X-ray)
C: Dynamite
D: Denwa (telephone)
Q14 <<¥750,000>>
In the anime Peter Pan, how old is Peter Pan?
(choices self-explanatory)
I'm not entirely sure to which anime they're referring.
Q15 <<¥1 MILLION>>
The kanji for "bug" (mushi) was originally a pictogram for what creature?
A: Koganemushi (Scarab)
B: Nezumi (Rat)
C: Hebi (Snake)
D: Kaeru (Frog)
The STUDY video after the fact shows the original form of the "mushi" kanji from (I think) 3,000 years ago, which looked a bit more like a snake. The one that shows a ladybug (three of that same character lumped together) doesn't mean ladybug specifically, just any generic worm or insect.
Wiki seems to back me up on that, but I may still have a few facts wrong due to translation.