Quoted from Isochronic_Frost:

Firstly, welcome to Pinside! I see you're a fairly new poster. Try finishing your profile, it really helps around here to have a finished profile!
Now. I own and love Baby Pac-Man and have thought or looked into all of these at one time. The answers will disappoint so be ready.
I'm going to be blatantly honest with you here, please don't be offended (too much). I'm going to use caps since it's easier to read
1) Can and older lamp set up (see picture) be converted into the newer set up where the lights are all screwed onto a board?

No, it cannot
-what will I need

You would need to swap in the entire new playfield with the circuit board/lamp matrix
-is it more trouble then its worth

absolutely 100000% It is not worth the time and it is extremely tedious
-is the newer light board even compatible with the older set up

No, the wiring is completely different so the playfield will have to be swapped for the lamp matrix one
-where do I find the parts needed to make the conversion
-if I can do the conversion Ill be purchasing a new overlay from phoenix arcade (even though my original playfield is an 8/10 condition)

The overlay from Phoenix Arcade is honestly crap and a waste of your time. Don't bother if you playfield is really that nice 8/10 is pretty damn good, this game was beaten to hell in the arcade (if they could keep it working)
2) Where can I buy the wood portion of the playfield to stick the new overlay on?

This doesn't make much sense, I'm going to assume you mean the actual playfield itself. The entire playfield is made of wood. It is not a part, it is the entire thing. You cannot buy a new one, that is the entire reason an overlay exists. You either keep the original or put on a crappy overlay. However CPR is trying to get enough preorders to make a run of BPM (Baby Pac-Man) playfields. Put your name on the list and we will get closer to new playfields being made.
-Ill be using parts from the original playfield where I can so same thickness will be needed

This doesn't make sense either, the playfield is one solid piece of wood, you cannot simple change it. It is not easy, or fast, or simple. It is so involved that there is zero point in doing this until the reproductions are finally made.
3) How could I install lights under the pac man and ms pacman plastics (left and right of the flippers)
-what tools will I need
-where do I connect them to

Now this is actually possible! This mod exists and is popular, I have not done it yet but I plan to soon. Most people hook it off one of the GI (General Illumination) lights, the ones that are always turned on and light the playfield. There is a guide here on pinside.
4) crazy idea but would it be hard to modify the new playfield to use functioning slingshots or will this ruin the game?
-if I did this how will I install it and what if anything extra will I need
-where would it draw its power from

This will ruin the game. There were 5 small prototypes made (Hence the name Baby Pac-Man) and I believe those 5 had active slingshots. They were so chaotic and hard to control they were changed to dead slings. The dead slings are bouncy enough since the game is only half sized! If you still want more action in the slings, consider changing it to superbandz or titan rings.