How to outline vector files in Illustrator

Hi,

I'm relatively new to Illustrator (using CS6). I'm working with some vector files I downloaded from Freepik and trying to create shapes I can cut out with a laser cutter. To cut the laser requires a file where the outline is continuous. Any gaps/interruptions in the outline will prevent the laser from fully cutting out the file.  So... in essence, I want to know how to take a preexisting vector file (not create a vector using image trace) and create an outline of it.

Thus far, I've had two main problems:

1. Situations where there are duplicate outlines/strokes

I've had mixed success in using the stroke feature. For example, in the frame closest to the word elements​, I tried turning on the stroke (shown below in red), only to find out that the outer most line (my outline) has a stroke on either side and a fill in the center. If I use the stroke here with the laser cutter, the laser will literally cut out the shape twice, once along each stroke. This leaves me with two cutouts. One of the interior white shape and one of the gray fill between the two red strokes. This isn't the end of the world, but it does make it harder to measure shapes and makes the cutting process take longer than it has to. Is there a way to either ditch the inner stroke or easily create a new outline of this shape?

2. Situations where there is no unbroken outline

These dog images are made up of a bunch of different shapes and don't have a continuous outline. If I try to use the stroke feature here (in red) the laser wouldn't cut a solid outline of a dog, it would cut a bunch of little dog segments. As above, is there a relatively easy way to create an outline of this vector?

Any help is much appreciated!

Thanks!