1. With which DoH class do I start?
- Depends on which you want to achieve first :
Carpenter lets you HQ easier with Byregot's Blessing crossclass skill at Level 50
Blacksmith lets you craft a variety of weapons and also many other crafter's main/offhand tools
Armorer's skill is usually needed in later levels, also lets you craft tank gears and also some crafter tools
Goldsmith lets you craft accessories
Leatherworker and Weaver both have gears necessary for your crafters. They also have necessary crossclass skills to help ease crafting a lot.
Alchemist has CP restoration crossclass skill that is good for early levels.
Culinarian for food buffs.

2. How do I level up with a decent pace?
Once you unlock levequests, you can start doing turn-ins for EXP. Also Grand Company allows you to turn-in one specific item requested at your level for plenty of EXP once per day for each crafter class. Completing challenge log also yields as much but once per week.

3. What's the use of crafting?
Having all crafters at high level has a benefit. Aside from able to craft things on your own for cheaper cost and make money, you can also repair gears on your own using Dark Matters. Unlike repairing with NPC, you get to repair items pass their 100% durability up to 199% so they'll last longer.