Answers to your first two questions.
in 2D, Right click on yourself to bring up your context menu (the same one where the insurance options are on non-SP shards). When you are doing one of the "Classic quests" (the Naturalist quest, the solen quests, the Collector quest and the hag quest - also the new character quests from Haven & Umbra, from before New Haven), the quest shows up as "View Quest Log" and "Abandon Quest" in the context menu. "View Quest log" will let you see what quest is active, and what is needed for the quest (such as the creatures the tinker wants you to take pics of). The "Abandon Quest" option kills the quest, without confirmation IIRC.
You can only have one classic quest at a time - if you have another one open, the NPCs for the others will tell you that you are too busy. The most common occurance of this are people who never finished the new Paladin quest from Haven, or were doing the Hag quest, and had to stop because of needing to make a star map (as you can only do the latter every other hour, from 5PM to 5 AM in-game time).
And, for the last one:
As for the statue piece, 10 of them can be combined into an obsidian statue that typically sell for 25-100k, depending on which of the 50 or so random names pop up on them. One of them, the rarest, occurs with the name of the person doing the assembling, and is worth MUCH, MUCH, more (but are rarely sold due to their personal value)