The issue is whether the E card can test the hypothesis that the rule of vowel-even is true. To test means to allow for possibilities which are consistent with the rule or which are inconsistent with it. The E card can be in either of 2 types: an even number or an odd number (and not just 4 or 9). The even number would be consistent with the rule but the odd number would disprove it. It is this possibility which makes the E card is a valid test. Note that if the E card had an even number on it, this does not prove the rule. The rule must be shown to work in all 4 cases in order to be proven. This is what I think you are missing in your logic. The problem is about testing the rule, not proving it. If the test is to be valid then it must be possible to disprove the rule. Validity of the results is not the issue. Validity of the test is the issue.
The result "4c" is not meaningless. It is consistent with the rule. The result 4a is also consistent with the rule. It is the fact that both possibilities are consistent with the rule that makes this card invalid as a test. Because neither of the outcomes disproves the rule. The 4 card can also return a lot of other results such as 4j, 4o, 4z and so on, which you seem to have ignored.
Again, the result 7c is not meaningless. It is consistent with the rule. The result 7a is a possibility and this possibility would disprove the rule. Because one of the outcomes would disprove the rule, this card is a valid test of the rule. And again, there are other possible results which you have ignored.
Again, these results are not meaningless. They are consistent with the rule. There are also other possible results but because all of them would be consistent with the rule (as with the 4 card) then it cannot test the rule.
You seem to have confused provability with testability. Proving a rule is a great deal harder than disproving it and is why almost no scientific theories are claimed to be proven.
There is another issue of logic here which is bothersome. In what I said above, and in your list of permutations, we are assuming that there are only 4 cards in total and that the rule only applies to those 4 cards. You are further assuming that only certain numbers and certain letters are allowed. However, we are only presented with one side of 4 cards, not with their reverse sides. The only assumption we need to make is (as you said) that a card must have a letter on one side and a number on the other side.
Then, if the the number of cards was infinite, we could never prove the rule, though we could easily disprove it. If the number of cards was just the four posed in the problem, or some other finite number, then we could prove the rule by examining the backs of all cards. We don't necessarily need to examine the backs of all cards to disprove the rule. We just need to find the first instance of a card that is inconsistent with the rule. That is why we look at the E and the 7 to test this. We are not asked to prove the rule.
Finally, if you stated the problem slightly differently and said that the test should only consist of cards with consonants on the letter side, then the rule is self-proving and cannot be tested. All the cards would by definition be consistent with the rule so the rule would be proved. But the rule cannot be tested in this way. This is what (supposedly) distinguishes a scientific theory from a non-scientific one.
