Regarding your example, a tree diagram might help to explain it properly. Also, basic stochastics often comes down to counting problems, that is we have to rephrase the problem such that every event gets the same probability (in your example, we are looking for four eight events out of 52, all having the same probability). If you can make your students understand this, you will help them a lot.
For an other example, take the question: "Throwing two dices, what is the chance of having the same number on each of them?". To properly solve it and more importantly develop the corresponding theory, one should look at all $36$ possible outcomes and from there deduce that the chance is $6/36 = 1/6$. There are of course other ways to see it (e.g. "fix one dice, what is the chance that the other gives the same?"), but in my opinion, turning it into a Laplace experiment is the one that can be used most often in such problems.
If you have control over the exercises, make your students always write down the total set of possible outcomes (maybe paraphrased if too big) and the set they are interested in.
Btw, a little advice on your way of explaining it with the probability bigger than one: This helps the student to see that there is something wrong there. Depending on their level of understanding, they will either say "ok, stupid rule with the one, whatever" or they will agree and understand that that can not be, however, they will still not be any wiser. You can use such an example with a very intelligent student to show him that he is wrong, but the general "I fu***** hate math!" student might just get demotivated by "BAM, you are wrong!", so a constructive approach to show them how to properly do it might be better. (I know I am exaggerating here and you most likely don't do it like that.^^)
A nice thing my own high school teacher did was the following:
Every Friday, he gave us two different experiments (with dices, drawing balls,...). Then he left the room and gave us about 10min to discuss them, we where to compute the probability. Then he came back and we did the experiment we, as a class, chose. If we "won" - that is, we succeeded for example to through two times the same number with dices, we got no homework for the weekend. The chances for success where always between 10% and 30%, so we really had to figure out which one to choose if we wanted good chances. :)
Of course for something like that, you need to know your class, you need the people to be motivated and you might not want to leave 30+ students alone to discuss (in this case maybe a majority vote for one of the systems might be better), but having such active experiments might always help.
Another nice thing he did (now I am really starting to remember^^): In the very first class, he wanted to show us that we had a natural understanding for Laplace experiments, all outcomes having the same probability. So he brought a dice, said we will through it 100 times and we should assume how often we will see a one. Turns out, we had 92x a one in the end, because it was a wooden dice and he filled the six holes opposite to the one with some heavy metal... xD
However, we afterwards discussed why we, without knowing anything about the dice, assumed to see about $100/6$ ones and this was a great and fun start into the topic.
But to come back to your question of other examples: I think the most difficult thing is to find the right model for a problem. Most errors I myself made or encountered were of the form "damn, I should have counted that without repetition, with order,...". So I claim that the most difficult thing in basic stochastics is not to do counting, not to compute binomial coefficients but rather to translate an exercise into a model. It is also often not clear from the text if they are considering repetitions, etc., so you might want to make sure that your students not only write down computations or results but also a little explanation on which model they chose and why.