To confirm your answer to the spreadsheet problem, make a copy of the grade distribution worksheet, and copy your "copy/paste" and "copy/sorted" columns into the appropriate places, then use the "recalculate" command if needed. You will have to update the mode and median by hand.
Q4. Why doesn't it make sense to find the mean?
Letter grades are not numbers, so adding and subtracting them doesn't make sense. Furthermore there are at least 3 sensible ways to convert letter grades to numbers (using the student's original score, the 4-point scale, and a 100-point scale adapted to the range of each grade such as the midpoint), so you can't even argue that there is an obvious correspondence to numbers. On the other hand, you can count the frequency of each grade, so the mode makes sense, and you can order the grades and take the middle one, so the median makes sense.
Q6. Is there a difference between the mean computed in Q2 and the mean computed in Q5?
To make sense of this question, we need to convert the two means to the same scale. The easiest way is to take the mean of scores from Q2 and convert to a letter grade, and then to the 4 point scale. This results in an integer, while the mean of letter grades will usually be a fraction. So there will be a difference, in this method because converting to a letter grade transforms a continuous variable to a discrete one, while take the average of integers does the opposite.
Other ways of converting will lead to different kinds of differences.
Whether this difference is "significant" depends on your exact data. For example, suppose the exact scores were 81 and 31, giving an exact mean of 56, or a D = 1.0. But in terms of grade points that's A = 4.0 and D = 1.0, for an average of 2.5.
Q7. What is the type of the "is A?" and similar variables?
These are a special type of variable called a dummy variable.
In more general terms we can classify them as discrete, unordered, non-cardinal variables. Conventionally we choose "yes" = 1 and "no" = 0, but can we say that "yes" is "bigger" or "better" than "no"? Perhaps if "is A?" is "yes", that is better than "no", but surely the opposite is true for "is D?", and it's not clear how to think about "is B?" and "is C?"! So these are not ordinal.
Nor does it make much sense to say "how different is 'yes' from 'no'?" They're completely different, and there is no "maybe" to compare to. So we can't say there is a "unit" for measuring how different two values are.