Looks right.

Cover the statements. Anticipate the sufficient info. This is a yes/no question. It's sufficient if it reveals a yes or a no.

Statements, alone or together, must limit the number of even integers to a finite number (yes), or reveal that there can't be only a finite number of even integers (no).

Reveal statement 1:

(1) The first ten integers in S are even.

What about the next 100, or the next 1000? How about a rule for the sequence? This is effectively 0 information. The first ten might be the only even numbers (yes answer), or the first ten and every other number after the 10th might be even (no answer). Insufficient. Eliminate A and D. Only B, C, E remain.

Reveal statement 2 after hitting yourself in the head with a brick so you forget statement 1.

(2) An infinite number of integers in S are odd.

Ok, but are all the numbers odd (yes answer)? Or every other number can be odd, still leaving an infinite number of even numbers (no answer). Eliminate B. Only C and E remain.

1 and 2 together - First ten are even, and an infinite number are odd.

An infinite number, (the first ten and an infinite number more), can still be even (no answer), or only those first ten are even (yes answer). Eliminate C.

E

You didn't include an explanation for why C isn't the answer. Find an exception just to make sure.