Thanks henelol. The turn-in checks for these were already counting correctly, but the Quest Log never filled in their progress lines, so they looked stuck at 0. Fixed: Quest Specialist now shows your completed quests out of 800 with its gauge (you're at 315), the Explorer titles count the maps you've explored in their region (Beginner Explorer is 13 of 20 Victoria maps for you), and Diligent Explorer shows its time left, today's play time out of 3 hours, and your qualifying days out of 27. PQ Mania is the one still open: it needs Party Quest S ranks, and the game doesn't track PQ ranks yet, so it can't be completed for now. That one is a design decision for the owner. Re-test: open the Quest Log (Q) and select each title.

Update: PQ Mania can now be completed. Party quests now keep a record per PQ (runs entered, runs cleared, your fastest clear, and wins for the Monster Carnival), and each record gets a rank from S to F by the original game's rules for that PQ. For S you generally need around 100 runs, a high clear rate, a fast best time, and that PQ's reward item in your inventory. PQ Mania asks for rank S in 5 party quests, and the Quest Log shows your count out of 5. Your record starts counting from now: runs before today weren't recorded.