1. In previous expansions it was a race at the start to reach max level. In preparation I would complete a bunch of quests and wait to turn them in when the expansion went live. Would it still make sense to do that? I think there are like 8 quests at the Lion's Den, each awarding 15k XP, for example.

I don't think it really makes a difference. The time it takes to travel and turn in these quests could be better spent starting the new content.

2. If racing to reach 110 isn't really the focus in Legion, what is? Finding and enhancing your artifact weapon? Reaching Dalaran and establishing rep with your class order hall? Leveling your professions?

While you need to level to 110, it's not too much of a race, since raids do not open until a week or two after the release of the expansion. The quest progression will naturally take you through the introductory scenario, reaching your class order hall and Dalaran, and acquiring your first artifact weapon. Leveling your professions and acquiring recipes requires traveling throughout the Broken Isles and can be done while you're leveling through those zones. Artifact power is gained throughout questing, dungeons, and world quests once you reach 110.

3. Does it make sense to do the Broken Isles zones in any particular order or is the order dictated by the quests to get the artifact weapon specific to your class and spec?

You can do the four Broken Isles leveling zones in any order - It doesn't matter. You can even go back and forth between them if you'd like. Most of the mobs scale to your level, even if you're grouped with players of different levels. The exception is the max level zone, Suramar. The quests to get your artifact weapon(s) take place before the Broken Isles campaign and take place in many different areas of the world, not just the Broken Isles.

4. From what I've read, your crafting professions will allow you to equip your characters with a full set of level 815 gear. Beyond that, they're on their own - to upgrade the crafted gear to 850 they will have to obtain Blood of Sargeras and Obliterum. Is this a viable means of upgrading your characters, or will they quickly get better gear from questing, dungeons, world quests, etc.

I haven't done a lot of crafting, but you will be able to equip as many crafted items as you can get - no more 3 item limit. The secondary stats are random, but can't be rerolled. Instead, unwanted gear can be turned into Obliterum, which costs Blood of Sargeras. Blood of Sargeras is BoP, and can be obtained by gathering, but also is dropped by bosses in dungeons and rewarded by profession related world quests. Obliterum will increase the item level of crafted gear, but the forge requires a basic piece of crafted gear of each armor type to unlock.

Crafting gear is a good way to gear up if you have the materials or gold, but it is not required to craft a bunch of gear. After reaching level 110 you will be able to do world quests and world bosses that reward gear that have a chance to be upgraded to a higher item level. The minimum item level of gear rewarded from world quests also scale with your item level.
Dungeons will also of course drop gear. Item level 810 is required for Heroic dungeons. Further, Mythic dungeons will reward better gear and will be available at launch. Mythic Keystone dungeons will be unlocked the same time as raids. Also the class order halls offer a set of gear that unlocked in various ways like progressing through the class campaign and obtaining order hall resources.

5. Is there anything that we can do to prepare for the Legion release (other than perhaps saving quests to turn in)?

You could save all your salvage crates so that the appearance of the gear in them will be automatically added to the new wardrobe system when opened. Keep in the mind the vendor price of follower equipment will be drastically reduced, so if you care more about gold you should sell those now. Garrison gold missions will be removed after Legion. The moose mount will no longer be available after patch 7.0. You could craft flasks and potions to increase your damage and prevent deaths while questing. I highly recommend getting a water strider mount to walk on water since there is a lot of water areas in the Broken Isles. The one from the Anglers in Pandaria is easier to get.