[Note:  There may well be alternate answers that are perfectly acceptable (particularly when it comes to the questions of fallacies).  If you think you have an alternative correct answer to any of these questions, check it with me.]

1. Express this argument in premises and conclusion form (including any missing premises or conclusion).

The most important task facing educators today is teaching students how to write well.  Those who write well will be able to understand the work of our culture's great thinkers.  Better still, those who can write well will be able to communicate effectively in today’s world.

Solution:  P1: Those who write well will be able to understand the work of our culture's great thinkers.
P2: Those who can write well will be able to communicate effectively in today’s world.
MP3:  The main purposes of education are to enable students to understand their culture and, most importantly, to communicate effectively in today's world.
C: The most important task facing educators today is teaching students how to write well.

[Note:  Here, there are plenty of other possibilities for the missing premise.  What's important is that there be some premise which makes it clear how we get from P1 and P2 to our conclusion.]

P1: The Liberals won more seats in the last election than any other party.
C: The Liberals received more votes than any other party in the last election.

Solution:  No.  It is possible to have P1 be true while the conclusion is false.  For instance, suppose the Liberals won 201 seats while the P.C.'s won 200 seats (and, for simplicity, assume no one else won any seats).  But even if this is true, it's still possible that the Liberals won each of their seats by 4 votes, while the P.C.'s won each of their seats by 3,000 votes.  In other words, it's quite possible that more people voted P.C. than Liberal.  In this example, the premises of the argument are true (i.e., P1 is true) but the conclusion is false.  But an argument can't be valid if it's possible to have all true premises and a false conclusion, so the argument isn't deductively valid.

3. The following claim is true:

An argument whose conclusion is a tautology is a deductively valid argument.

Solution:  P1: A deductively valid argument is one which cannot have all its premises true while its conclusion is false.
P2: If an argument has a tautology for a conclusion, its conclusion cannot be false.
P3: If an argument's conclusion cannot be false, then the argument's premises cannot all be true while its conclusion is false.
C: An argument whose conclusion is a tautology is a deductively valid argument.

[Note: There are other ways of answering this question.  Here's a tricky way:

P1: An argument whose conclusion is a tautology is a deductively valid argument.
C: An argument whose conclusion is a tautology is a deductively valid argument.

Although, this clearly begs the question, it is sound so it's a perfectly good answer to the question.]

4. Suppose all you know about an argument is that all its premises are true and its conclusion is true.  Can you conclude anything based on this about whether the argument is logically strong?

Solution:  No.  What you need to consider is whether you can tell anything about whether the argument is inductively strong or deductively valid (since these are the two ways of being logically strong).  Start with inductive strength.  Since you don't know what the premises and conclusion are, you can't tell whether the conclusion is likely to be true if the premises are true.  So, you can't tell anything about whether it's inductively strong or not.  What about deductive validity?  Here, you can't tell anything either.  The argument could be:

P1:  Roger Grimes is the leader of Newfoundland's Liberal Party.
C: It's February.

Probably, both P1 & C will be true when you read this argument, but they won't always be true.  In particular, about a month after I write this P1 will be true while C will be false.  So, even though the premises and conclusion are true now, it's not impossible to have true premises and a false conclusion.  In other words, the argument could be invalid.  On the other hand, the argument could be something like:

P1: It's February.
C: It's February.

If so, the argument is valid.  In other words, based on what you're told you can't tell anything about whether the argument is valid or not.

All in all, based on what you're told you can't tell whether the argument is logically strong or not.

5.  That sauce I ate must have aged cheese in it.  Aged cheese always gives me a terrible headache and right now I feel as if my head is going to explode.

This is a case of the fallacy of affirming the consequent.  The argument here is:

P1: If I eat aged cheese, then I get a terrible headache.
P2: I have a terrible headache.
C: I ate aged cheese.

As we've seen, any argument with this form commits the fallacy of affirming the consequent.

Solution:  Assuming Lincoln is serious, he's concluding that being addicted to whiskey is what causes Grant to be a successful general based on the fact that, in Grant's case, being a successful general is correlated with being addicted to whiskey.  Thus, Lincoln would be committing a causal fallacy.

Solution: This is a case of the ad hominem fallacy.  All the letter writer does is attack the professor, not what the professor is saying.

Solution:  This is a case of the straw man fallacy.  People who think we should reduce the prison population are misrepresented as thinking that "murderers and bank robbers should be free to commit their crimes without doing prison time" and that we should "turn ... vicious criminals loose with little or no punishment".  Surely, this isn't what those people actually think (typically, people making this sort of argument argue that we shouldn't send people to jail for non-violent crimes).

[Note:  there's an element of ad hominem here as well, but it's not actually true that the writer of the passage above only attacks the arguer, not the argument.  In addition to calling his opponents morons, he does present an argument against his opponents.  Unfortunately though, the argument he presents is a bad one because it commits the straw man fallacy.

In case you're wondering, I certainly wouldn't take any points off if you said ad hominem as well as straw man.]