What I've learned from experiments.

Weapon Damage

The level difference seems to have some impact. The high level character will not insta-kill most mobs on a low-level map. Evidence suggests that the weapon damage of the higher level character is scaled down. Specifically, I can visit old russia on earth with my level 20 character and still not one-shot everything. They go down easier than I remember, but some mobs may still require 2 shots.

Player Damage

It seems mobs also have a way of damaging you. If I run into a group of level 2 mobs on old russia as my level 20, I will still take damage enough to kill me. It seems to take longer than I remember but I'm not safe.

Experience

The level 1 character will earn XP if he's close enough (distance-wise) to the level 10 character. He will earn the same XP whether he kills the mob himself, or if it's killed by the level 10 character. Dregs gave us 20 xp each whether we helped kill it or not.

Specifically we tested with the lower level character just running alongside me and when I killed mobs, he got XP. Same way the other way around, if he killed a mob, I got XP even though I did nothing except tag along.

The same XP is awarded to both players, so obviously the higher level character will require a lot more dregs to be killed before levelling up.

Note the distance remark above. If the two players venture too far apart, no XP is rewarded from kills the other player does.

Item Drops

Item drops will mostly be according to the zone level. If you're playing as a level 10 character in a level 2 zone, you'll get level 2-3 drops, with the occasional item more appropriate for your level. Encrypted engrams will, as noted elsewhere on the interwebs, be decrypted at the level you have when you decrypt them so they're just as useful in a level 1 zone as a level 10 zone. No idea whether drop rate is affected though.

Item drops are also personal, which means that all the ammo and item bulbs that you see on the ground is yours.

Mission Rewards

Mission rewards is a one-time event. You only get the rewards when you complete the mission the first time. However, as a higher level character you have a higher chance to get encrypted or decoherent engrams containing shiny loot, which offsets this. It is not a waste of time to play with your lower level friends!

So to recap:

1. Low-level character earns XP when he kills mobs, and he earns the same XP when the high-level character kills mobs, the latter only if the two characters are actually playing together (ie. not on opposite ends of the map). I don't know what the distance range is but it wasn't exactly stand-on-top-of-each-other either.
2. Low-level character earns item drops as usual, according to the zone. As does the high-level character, but he'll get some item drops that aren't useful (too low level).
3. High-level character is slightly overpowered for the zone, but not to the extent that you can level the zone just by sneezing.

Conclusion

It's probably best to be fairly close in levels, then both/all players will have the most fun and the most to gain, but there seems to be no adverse effects of bringing a high-level friend to help a low-level character, except for the high-level character not really getting a lot out of the process except the friendship.

Mega-conclusion

Destiny is meant to be played together, it seems Bungie has made a system that allows this, even if the players are not the same level.

So team up soldiers, the darkness must be pushed back!

They are completely random. Typically they are offered when turning in other bounties. To increase your chances, keep up with your bounties!

Here is a reddit discussion about it. Not a perfect source, but there seems to be no indication that they are trigged by something specific.