How much damage do vehicle weapons vs. personal size chars

1) Correct, as long as a Natural Weapon is used for the attack. Without a natural weapon things would have been different. Also no Astral Combat skill is needed, if a natural weapon is used.

p. 399, Natural Weapon critter power

A dual-natured critter with a melee Natural Weapon can use this power against astral targets that are within its reach. Use the critter’s normal Unarmed Combat skill and physical Damage Value for this attack.

2) You're wrong. Dodging is done using Logic+Intuition; "soaking" means reducing the damage after a successful hit. Since the mage is only astrally present, Willpower (see table p. 314) is used to soak damage. (Manifesting doesn't make the mage vulnerable to attacks that wouldn't hurt her, if she would project without manifesting, so it's still astral combat)

p. 315 Astral Combat

Astral combat is resolved in the same way as physical combat.

p.314 Manifesting

Manifesting is a psychic effect that lets you make yourself visible and audible on the physical plane through an act of will. [...]

You can’t interact with anything physically (or be harmed by physical attacks). [...]

You’re still on the astral plane, so astral attacks can still target you.

4) Even if the mage would not be dead immediately, filling all boxes would basically kill her, since finding the body (p. 314, While you were out) would be impossible, if she is knocked out.

What could the mage have done?

1. This is Shadowrun. You don't just walk up to a potentially dangerous stranger and say "Hi"; you gather all info you can get, then maybe you talk. A better approach would be to stay in the astral and check that being using Assensing(assuming the skill rating is high enough to make it work with that INT attribute). Most paracritters shouldn't be too good at masking, so the mage would likely get the info that she is dealing with a dual natured being and would have approached it with more care. With the high move rate in the astral evading it shouldn't be too difficult unless the mage is in meele range and surprised.
2. What about burning edge? Not Dead Yet (p. 57) could have saved her.

Well, think about it like this, they are running away, and you wish someone on your troop had bought heavy weapons, but they didnt, at least we can try to score a huge success on our attacks and maybe hit some critical component on their ship, right?

Yes, the rules assume that the scale between personal and vehicle is 1:10, each 1 point of hull is about 10 wounds on personal scale, and 1 point of armor on a vehicle is about 10 points on personal scale. That is an optional rule that your table can ignore if you want to. But assuming the falcon is a regular corellian cruiser (which it isn't), they would have to hit it for about 31 points of damage to be able to even scratch it.

This conversion works fine if you are shooting at speeders or other vehicles with armor 1 or 2, otherwise it becomes almost impossible to damage the vehicle using simple blasters.

Playing on a smugglers game, we had our armor 3 customized yt1000 be damaged for 29 points of damage from a heavy repeating blaster (base 15 damage) operated by a rival trooper. Had he another point of armor piercing or two and extra successes he could have scratched it. But otherwise, you need weapons with the breach quality, like missiles and rockets.

Keep in mind that even if he did a single point of damage, it would be on personal scale, not vehicle, and would represent 1/10 of a hull threshold damage (or hull trauma), but already enough to trigger a critical component failure on a critical hit.

The strongest personal scale weapon i know of is the Verpine Heavy Shatter Rifle, that has 15 base damage and pierce 6, with accurate 2 on top of it, with enough hard points to add an augmented spin barrel. A character with 6 agility and 6 gunnery can easily hit for 12+ successes regularly using it. Having critical 2, it means it only needs 2 advantages to cause a critical hit, which can be reduced to a single advantage with the superior quality.

The Missile Tube has the breach 1 quality (ignoring 1 point of armor) and hits for at least 20 damage (plus successes). While the Flechette Launcher has breach 2 (ignoring 2 points of vehicle armor) and deals 10 damage. But these are explosives and designed to be used against structures or vehicles.