Question 1. I would like John and me/I to continue sharing them.
At least you didn’t commit the common error of using “myself”. Let’s dissect this sentence. The main verb is “like”. The subject of the sentence is who is doing the liking; therefore, the subject is “I”. The object of the liking is the sharing. Imagine the object of the verb is something that you could do on your own, like winning the lottery. Could you say “I would like I to win the lottery”? Clearly the answer would be “I would like me to win the lottery.”
This is like the question of who and whom. Remember we use whom when it is the object of the main verb, even if it is the subject of a secondary verb.
But the best way is to rephrase the sentence “I would like to continue sharing them with John”
Question 2. I am in a perpetual or perpetually bad mood.
Since “perpetual” modifies “mood” it’s an adjective; use “perpetual”. “Perpetually” would have to modify a verb. I’m perpetually running out of patience. I guess you could argue that “perpetually” modifies “am” in the sense that the state of being is ongoing. But to support that meaning we would change the syntax, to “I am perpetually in a bad mood.”
Question 3. We need time for us/ourselves.
Since this is a reflexive sense, the correct pronoun is ourselves.
Question 4. Is/are three pairs of pants enough?
Are three cups of coffee enough? Don’t get confused by the apparent pluralism of “pairs of pants.” The key factor is “three”. But a tougher question is: What if it were one pair of pants? A: Is one pair of pants enough? I guess it depends on where you are going.