Well, as the title suggests, I've never modded before,  but I would like to do some light mod work to rebalance the game for fun. So, I have a few questions before I start:

1. Can you change a ship's abilities to be another ship's abilities without changing the second ship's abilities? My current idea is, because I am a strident Advent player and because I'd like to do some testing, to replace the Radiance Battleship's Detonate Antimatter and Anomosity with Meteor Strike and Mass Disorientation, respectively, in three level forms. As a side question, though I'd imagine this is possible I'll ask anyway, could I increase the main and bleed off power of Cleansing Brilliance?

For the first part you can change the abilities that an entity has by opening the entity's file. the secound part is also possible, but you will also have to edit the abilities so that the level of the ability increse with the capital ability instead of by upgrades.

There is a lot of things to learn (it is simple, but there still is alot) in generel and I recommend that you start of by opening the different files (ships, abilities, buffs etc.) and see how they are "build". The content is usually named in a way so it is easy to understand what it does.
I also recommend that you read the basic guide that follow with forgetools 3. Some of the content is outdated, but it is a good introduction, the reference files are also outdated, so don't use them.

2. Could I let races have more than just their five capital ships, and/or switch out the ones they currently have. I have plans for all three races, but as an example, I'd like the Advent to have, in addition to the Radiance, Halcyon, and Rapture, the Akkan, Kol, Kortul, Desolator, Marza, and Skirantra.

You can edit that in the race entity files. However you can only have a max of 9 capital ships.

3. How would I go about changing which way a ship faced to engage? i.e. Increase the damage of the Dunov's missiles and autocannons, and equalize the laser damage, then make the defaut fire behavior to fire forward?

You can edit that in its mesh file (have to convert it to a txt file first). In the start of the mesh file there will be some points you then have to edit the orientation of the weapon points. Can't remember what number weapons the two are, but weapon-0 is the first weapon in the ships entity file and then so on.

the direction works this way:

Front:

    Point
        DataString "Weapon-1"
        Position [ 88.124916 14.478189 85.954254 ]
        Orientation
             [ 1.000000 0.000000 0.000000 ]
             [ 0.000000 1.000000 0.000000 ]
             [ 0.000000 0.000000 1.000000 ]

Back:

    Point
        DataString "Weapon-1"
        Position [ 88.124916 14.478189 85.954254 ]
        Orientation
             [ -1.000000 0.000000 0.000000 ]
             [ 0.000000 1.000000 0.000000 ]
             [ 0.000000 0.000000 -1.000000 ]

Left:

    Point
        DataString "Weapon-1"
        Position [ 88.124916 14.478189 85.954254 ]
        Orientation
             [ 0.000000 0.000000 -1.000000 ]
             [ 0.000000 1.000000 0.000000 ]
             [ 1.000000 0.000000 0.000000 ]

Right:

    Point
        DataString "Weapon-1"
        Position [ 88.124916 14.478189 85.954254 ]
        Orientation
             [ 0.000000 0.000000 1.000000 ]
             [ 0.000000 1.000000 0.000000 ]
             [ -1.000000 0.000000 0.000000 ]

Thanks to Aractain for this info.  

4. Would it be possible to, (as I'm running Entrenchment), bring back the old fighters and bombers as alternate strike craft for their respective races? So, the old Advent Fighter and TEC Bomber could become interceptors, with light armor and weapons but in a large squadron the flies and fires extremely quickly? Or are those files not workable?

A squadron concist of two entity files. The first is the squadron entity file which sets how fast the fighters are build, how many fighters there is, how close they fly together, which fighter the squdron is made of (can only be one) and such things. Then there is the fighter entity file. This have the information for the individual fighter such as angular thrust (how fast it turns), linear thrust (how fast it moves forward), damage, weapon prebuff (how fast it fires/rate of fire) and so on.

5. Alter the firing arc of certain weapon turrets or origin points without changing the whole set? Like, on the Radiance, which has 10 laser turrets on each side, make the four turrets farthest forward able to fire forward and sideways, and the four back most able to fire sideways and backwards, but not all 10 able to fire forward and backwards in addition to sideways?

as you can see above there is only the option of front, back, right and left. You can change so that the points will face in different ways, but it will always target within one of these four.

6. Increase the fighter/bomber spawn rate? Not, say, instant, but so that it was 25-35% faster. I plan on reducing the durability of fighters and bombers so as to make flak frigates more reliable, but I don't want to nerf strike craft too badly, so I want them to be brought into battle faster.

This should have been answered above.

7. Give abilities to the two flak frigates that don't have them? This ties into the first question, but, I'd like to give the Garda and Defense Vessel a reduced version of Power Surge without the shield regen.

the abilities are in a different location of their entity file then the capital ships, but it works the same way as in question 1.

8. Make the Heavy Fighter passive on the Sova Carrier also make the fighters slower, or have less chance to dodge?

hmm, could you be a little more prcise on this one? For the dodge, there is a way to make an passive ability that you can add to the bombers which will decrease their change to dodge.

9. And lastly, make Embargo's denial of phase jump affect ALL ships in the local gravity well?

This can be done by edditing the targetfiler of the abillity and/or its buffs