Hey there, welcome to the forums!
1. When leveling hunting/mining/foraging I would not send the sprite training but collect materials, as you can use those for crafting or training your crafting skill/salvaging.
About leveling your crafting skill, you could train via crafting and if you are lucky to craft an orange weapon/gear you could sell it.
By doing that you would level and earn some money at once, I personally spam training though on crafting skills/salvaging, as I don't always keep all required mats with me.
Bow and arrow/firearm crafting is easily leveled by letting your sprite mass-produce arrows/bullets, at least at lower levels.
2. Auto activated quests up to level 30 are to help you understand the game, this comes in handy if you just joined or didn't quite grasp some functions yet.
Auto-activated quests past level 30 are usually dungeon quest chains or character quests such as unlocking the sun talents. I would recommend doing them, cause the blinking can get annoying or if you accept them your quest log gets clustered.
Additionally, if you want to enter a dungeon's endboss' room you need to have the quest chain for that dungeon finished, so I would recommend doing dungeon quests for sure at some point.
If you don't want to though, you can skip those. Though, you should do "character" quests (such as the sun talent unlock quest I previously mentioned).
3. When you open any sprite's cottage and click on 'alchemy' and change the tab to 'M-Alch' you can see the M-clay table.
Most items are indeed tradable, though, sometimes there are non-tradable items on the table. Simply hover over the items on the table, if they are not tradable then the item will state so.
In today's mclay table we have the 'Jackpot Ticket' for instance, this item is not tradable.