How to analyze the lower bound of the horse racing problem using adversary argument?
Intuitively, the reason you need $7$ comparisons is that (1) you need $5$ comparisons just to see all the horses, (2) you need to compare all the first ranked horses to see which one is the fastest, (3) at this point you can't tell which is the second ranked horse.

Of course, this is not quite an adversary argument, and some work will be needed, though it is easy to use an adversary argument to show that (1) is correct: suppose that the strategy involves less than $5$ comparisons; then you don't see all the horses; so you can't tell whether the fastest horse is one of those you've seen or not. (Formally, if the algorithm states that a horse it's seen is the fastest, then you arrange for one of the one it didn't see to be the fastest; and vice versa in the other case.)

The same argument can be extended to show that $5$ comparisons are not enough: the only way that $5$ comparisons could be enough is by executing step 2 of your algorithm, and then you cannot tell which horse is the fastest. (I'll let you formalize that.)

I'll let you complete the argument and show that $6$ comparisons are also not enough.