Let's look at the given fact that Malachi arrived on Friday (#5) and keep that in mind while we look at #7:
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7. Of the cat owner and the Hellboy fan, one is Eli and the other arrived on Monday.

Since (Malachi is not Eli) and (Malachi did not arrive on Monday), we know Malachi is neither the cat owner nor the Hellboy fan.
Let's do the same thing with #6:
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6. Of the Batman fan and Kylee, one arrived on Wednesday and the other owns a frog.

We know Malachi did not arrive on Wednesday, we know that Malachi does not own a frog, which satisfies both conditions on the right, so we know Malachi is neither the Batman fan nor Kylee (though that part is easy).
We have now eliminated every possibility for Malachi's fanness except Iron Man, so you have another green dot and some more Xs.
Let's look at #4:
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The Iron Man fan arrived the day after Christopher.

Malachi the Iron Man fan arrived on Friday, therefore, Christopher arrived on Thursday. Another green dot and more Xs.
Go back to #6 and look at the right side. Frog is Monday or Tuesday, the other clue is Wednesday, so X out the possibilities on the left for Thursday and Friday. This is just one new X, so Thursday is not the Batman fan.
Repeat that with #7. Same logic, so neither the cat owner nor the Hellboy fan is Thursday or Friday, so two more Xs and a green dot for Thursday being the Wolverine fan, which leads to more Xs.