Neither did "Encounter at Farpoint," "All Good Things...," or "Death Wish." Only 2/3 of the episodes featuring Q had "Q" in the title. The same goes for "All Good Things...". The same goes for "True Q" and "All Good Things...". The most you can legitimately say about "Tapestry" from these points is that it might not have really happened. It can't be conclusively proven that it did, but neither can it be conclusively proven that it didn't. Absence of evidence is not evidence of absence.
Picard mentions that there were untidy parts of him he wished he could remove. Q simply may have been the most logical and/or immediate paradigm his mind could generate to explore those regrets. The old you can't prove a negative. Picard is uncertain as to whether or not it was truly Q and the audience is left uncertain. This ambiguity is one of the things that makes this episode so wonderful. The other thing I love about this episode is that like the TOS episode "The Enemy Within" did with Kirk's character, "Tapestry" does a beautiful job of showing that it takes both the erudite and well as the "Hell bent for leather" ensign who was stabbed through the heart in a barroom brawl facets to make Picard the flagship captain. Warmest Wishes, Whoa Nellie
My memory may be faulty (and usually is), but I thought I recalled Q refering to humans as some kind of single celled life form in comparison to himself. Can't remember what ep, but if he did say it, it would give creedence to the "pet" theroy.
Don't know if Q love Picard, but more that he had weakness for the crew of the Enterprise and mankind. He treated them like a favourite toy that he would throw about when he was bored.