I'm not sure where I'm supposed to configure my tool settings, in my cam software or mach3, but my z-axis goes up when it should be going down (into the wood to make a cut), and I'm not sure what some of my settings should be.

Let me run you through my process:
1- I open lazycam, and open the .dxf file. 

2- I then clean it up using the following AutoClean options: "Optimise Drawing" and "Relayer" (because I have some inside cuts).

3- I apply tool #1 (as I have set) to the inside layer, and the outside later.
My Router tool has the following settings: (These were the settings I was given by the person I purchased the CNC from, I am NOT sure if they are correct, except for the RPMs on my router)
Spindle speed:35,000rpm
Per pass: .25units
Plunge Feed: 50 units/minute
FeedRate: 80 units/minute

I do have inches set as the definition of units in mach3 and lazycam.

4- I applied the above to all layers of the .lcam project, then exported the gcode.

5- I open Mach3, and load the gcode... if I set the tool info in lazycam, do I need to configure anything else for the z-axis to work fine?

Under the "layers" tab on lazycam, What are the following supposed to signify?
Rapid Height: what does this value mean?
Cut start: What should this value be?
Cut Depth: (I'm pretty sure this one is obvious, how deep I'm cutting)

Hypothetically, if I have everything under the layers tab (Tool and Cut) configured properly, I can post the g-code to mach3, and not have to configure anything more in mach3, correct?  Cutting should be ready to go as long as mach3 has the correct 0,0 coordinates?

Upon running the machine, it seems like the tool is going UPWARD instead of downward into the wood, and then it randomly does make it into the wood, but when it DOES, it doesn't go as deep as it should be, which makes me think I don't know jack about getting the tool and z-axis set properly.

I've included the current .dxf file, .LCam file, and .tap gcode file that I'm working with, if anyone can point out where my problem is...

I don't use LazyCAM, but looking at your code, it looks like your cutting depth is Z=0, and your rapid height is Z=1. So it really should just be skimming the surface, and not actually cutting anything.

do I need to configure anything else for the z-axis to work fine?

Yes, you need to set Cut while in layers as that defines the rapid height, where the cutting will start, and the cut depth ( which by the way should be a negative number).
See Tutorial #7 in the LC Manual. It you don't do the above it will use the defualts.

RICH

Well, cut depth not being negative sure explains a lot.

Tutorial 7 helped a lot as well, thanks again for the great support, Rich.