Which mage pet for soloing on green?
 

AFAIK it's either Earth for root nuking or Fire for the DS dps. When would you use what and why?

Water. Water is the absolute champ for levels 4-20 if we're talking about solo. You never want a rooted mob that would prevent the 'death march' of a dying mob if you are doing solo work outdoors. Water has a decent DD spell that matches Fire, which is 'meh' but better than Root for making something dead. Water also has the 2nd highest health, just below earth.

For grouping, it's Air elementals all day long, easy. The stun is just snazzy and always appreciated. I guess a case could be made for Earth in how it's constantly rooting and that would help a warrior style tank that has low aggro in Vanilla, assuming that he's the one standing nearest to the target. But Air is the one you could just use by default and everyone will love it.

Quick side note - absolutely no one is going to have access to Focused pets at the server's start, and that's a big deal for the solo magician. Keep in mind that pets can accept armor pieces and wear them, and substantially benefit from the AC. If you have the plat to spare and intend to do a solo camp for hours and are a bit cautious about your pet durability, even Tattered leather armor suits can make a difference.

At higher levels when I would solo certain dungeon camps where reclaiming a pet isn't such a hot idea, I'd cheerfully trash the under 30pp cost of a full set of banded armor (less one bracer) as I had the skill to make it and equip a max summon pet with it to make it far more durable. More than likely I'll do the same again on Green.

On Green it's gonna be whichever you can get from the vendor at each spell level. As you level you will often only be able to choose one and the rest will be research only. Fire is the best for reclaiming if you are able to find mobs that are just barely dark blue. They will melt on the DS and you finish them with one nuke. If you're grouping water is preferable (air vs. a lot of casters). If you're duoing Earth can be good, just keep him DSed and buffed with beefy HP buffs, he is very easy to keep up with a healer, even when tanking multiple mobs/breaking spawns.

Is pet exp penalty going to be active from day 1 on green? If so, how do you balance between dps for 51% damage and not taking aggro? (If you CAN even out-dps your own pet at earlier levels)

In my experience it's typically not worth out damaging your pet early on.

Fire Pet chain summoning from level 16 to 50
Earth pet for a personal tank
Water pet high level with backstab for grouping
Air pet optional to use against mobs that cast. It's stun can interrupt their casts sometimes.

Yes, even though it's not classic. Not classic at all.

Duo or small group with other pet classes. Your pets are going to be stronger than melee characters anyway. Couple encs, couple mages, couple necros... 6 pets or 5 pets + healer will be easymode. No pet xp penalty in groups.

6 pets or 5 pets is 2 too many or 1 too many.

How is chain summoning supposed to work?

Chain summoning just means summoning a new pet before it dies rather than letting it heal up. This doesn't really become a viable strategy in Vanilla EQ until pet costs decrease at level 29 and you can buy fire pet from a vendor. 34 - 39 is an awkward level range because you can't vendor fire pet, which is by far the best pet to chain summon because you don't need to waste mana putting a damage shield on it. You'll also be stuck with earth pet at level 49 if you don't find another mage who's willing to make a fire pet for you.