1. Tank specs are the easiest to level with, since you can deal moderate damage, but you will have a much greater up-time than a DPS spec due to your ability to handle more incoming damage. Healing specs are the most painful to level with unless you are constantly with a group and/or doing dungeons.
2. Depending on the professions, you can benefit from them while you level, both in terms of useful items and gold. However, at the same time, it is time consuming. If you're wanting to reach max level as fast as possible, you may want to pass up working on your professions. If you're in no rush, then by all means, go for it.
3. What the above two said. If the group you're with can handle the dungeon you're trying to do, it is often a good source of gear, and will be an excellent source of experience if you and your group are at the appropriate level for the dungeon.
@Your most recent post;
I'd recommend leveling as Shadow unless you're going to have a group with you 100% of the time. Shadow is capable of healing dungeons a decent amount up to a certain level threshold, which is where it will become too difficult to heal without proper healing talents. However, by the point that happens, you'll probably be high enough level to purchase dual-spec.