Why don't any animals have green fur? Wouldn't it be the best camouflage in many cases, like for tigers, etc.?

 The idea behind camouflage is to break up the body's silhouette. This makes it extremely difficult for prey to see a clear image of the predator. Humans also use this technique in times of war. A human example is shown below:

Please note that it is not a single color. It is mottled not only to blend in but also to break up the human form.

Now, look at the tiger below:

It simply blends into the scenery, its entire form broken up by the long grass. Humans have excellent daytime vision. Most animals cannot, and they have difficulty seeing shades of red in particular. The tiger blends into this background.

Here's another example with a leopard. This time, spots perfectly break up the animal's silhouette:

Prey animals also use camouflage to break up their silhouettes. It is difficult for a lion to keep its potential prey in sight in this crowd because the stripes all blend together:

The problem with green is that it's only useful in a specific environment, and shades of green are constantly changing. Spots and stripes are much better. Furthermore, prey animals see green much better than red.

Why is the Meenakshi goddess green?

 There’s no particular reason. She’s just green.

There are certainly some associations. She is sometimes associated with the goddess Matangi, who is also green. The color green is additionally associated with Budha, and is a symbol of intelligence or knowledge/wisdom. It’s generally an auspicious color, being the color of thriving, well-watered plantlife, which means abundant food and overall harmony in nature.

But ultimately, it’s simply that Bhaktas who have seen visions of the real goddess Meenakshi report that Her skin is green. That’s just what She looks like.