A path cannot be partially in front of and partially behind another object in the z-stacking order. You have to use multiple paths which just appear to be one object.
Basically, just cut an object where it doesn't overlap another and send one of the resulting objects forward/backward.
Photoshop's bevel effects are raster-based. There is no single-command equivalent in Illustrator. Other vector drawing programs can do bevel/emboss with rendering results similar to Photoshop's, but those also resort to raster effects.
Remember that the terms ‘in front’ and ‘behind’ only have limited meaning in a 2-dimensional space.
What you are talking about is only an illusion of ‘in front’ and ‘behind’.
To achieve this you will need to clip your objects (the letters) into pieces and arrange the pieces in the right stacking order (sub-layers) to achieve the illusion.
For this kind of work the Pathfinder panel is your friend; Divide and Unite are probably the most useful operations.
People often make this harder than it is. It's usually not necessary to using clipping paths or to make cuts confined to the areas of overlap to make objects appear to intertwine. Usually, you can simply cut where objects do not overlap. There are no masks, clipping paths, pathfinders, etc., involved in this knot:
Your specific situation may be somewhat complicated by the use of drop shadows, depending on the specific parameters.
Okay I got the overlapping done and everything done, but you still can't see it very well. Does anyone know how to help me with my problem that I mentioned in the PS, cause that might make it easier to see
Illustrator won't do any magic in 52 pixels x 103 pixels.
The image will be as bad as before, since pixels are pixels no matter which program created them.
Your design is simply not suitable for a small size like that.