"I would like it if you did not do that" -- why "did"?

You are right. The sentence is using the subjunctive mood. The conjugation of the subjunctive mood for the verb "to do" is "did" — not to be confused with the preterit, which is also "did." Sometimes such sentences become clearer examples of the subjunctive mood when you look at them uninverted, for example:

• If you did not do that, I would like it.

Another way to word it would be as follows:

• I would like it if you were to not do that.

To more aptly demonstrate the subjunctive mood in contrast to the preterit, a better example would use the first person singular rather than the second person, as follows:

• You would like it if I were to not do that.

You'll notice that the verb after "I" isn't "was." The first and third person singular conjugations of the verb "to be" are the only instances where the subjunctive mood doesn't exactly mimic the preterit tense in the words it applies.