1) How fast does the shield mitigation go up? Is it instaneous or would the first volley all get to count at only 15% shield mitigation? If it takes a second to kick in then the best thing would be to focus fire one volley into one target then change to focus fire a second volley into a second target and so on letting all focus fires hit at only 15% mitigation.
Mitigation increases at a rate of .1% per point of damage dealt. It decays at the rate of 1.25% per second (even if the ship is "in combat").
2) If left alone what will the fleet do by itself? Will it focus fire at all? Say i have 10 on 10 battle going, all ships are in range and are of the same type. Will this be 10 1 vs 1 battles?
Yes, as default ships of the same type prefer to focus fire. This can be changed in a mod. For example, usually most/all light frigates will shoot at the same target, long range frigates will shoot at the same target (that's also usually different from the light frigates' target) so on.
3) I assume the AI matches up damage and armor type to the most effiecent type if 2 mixed fleets are left alone. Correct?
Yes, and why mixed fleets focus fire on several targets at once. Ships attack ships of the armor type against which they get the most bonus damage. The table for this is in Gameplay.constants in GameInfo if you're curious.
4) How quickly does the shield mitigation go up? I mean for smaller vessel like a frigate, can it take enough damage that it even reaches maximum shield mitigation before it is destroyed?
See above, .1% per point of damage taken. Light frigates can definitely take enough damage to reach max mitigation, but it depends on how much incoming fire there is. It is possible to one shot a light frigate even through max mitigation 
5) What does this mean?
Shield Mitigation always decays at the same flat rate over time. A flat rate equivalent to the Shield Mitigation generated by 1.5 unupgraded Light Frigates.
Does it decay in battle at a worthwhile rate enough to alternate focus firing?
No idea what the quote is supposed to say, but at a 1.25% per second decrease, probably not. The other thing to consider is that all races have ships with various repairing abilities and all ships regenerate shields and hull even in combat, so waiting for mitigation to drop will give them time to get patched up with or without help.
6)Unrelated. If a carrier is destroyed, do its fighters and bombers live on to fight? What happens to them?
They start taking damage and eventually get destroyed. 1% per second sounds about right I think.