1. They stay in your journal. When you do your next set of leves, you'll be able to turn those completed quests in to "stack" your next one, which will increase it's reward.
Each leve has a point value that will show up next to it, and you can stack up to four leves. It's a straight multiplier, so you want to find the leve with the largest gil reward and stack as many points on that one as possible. Item rewards don't change yet, but I think they may in the future: after altering the reward, sometimes the item is also colored yellow to show that it's been modified.
Note that Local Levequests (crafting ones) do get cleared, because there is no stacking system for them, for some reason.
2. To make your own arrows, you'd primarily want Botanist for wood, Carpenter for working the wood, and Weaver for making fletchings. Arrowheads, however, look like they come from just about every other craft, but mostly Blacksmith and Goldsmith.
3. I'd just check ffxiv.yg.com and see which items you can use for one of the above crafts within 10 Ranks or so, and sell everything else.
4. Yeah, the dots indicate enemy difficulty. Blue is easiest, red is hardest. Goes like this:
Blue -> Green -> Yellow -> Orange -> Red
5. Under "Configuration" in the in-game menu, there is an option "Reset Widgets" at the bottom, which will snap everything back to their default positions.
Nobody can hack your account with just your login and password. It's an extra step when logging in, however. 
