I've never driven a car with lsd and I was wondering how I could check to see that its working while driving.

 

Go to a parking lot and "light up" the tires.
if 2 tires leave marks you got an LSD.
You can check also to see if there is a tag on the rear of the Differential housing.it says "LSD gear oil Only" on the fill plug.

 

All of my mounts are aluminum and it makes a loud rattling sound when I'm taking off. Is that normal?

 

Rattling is not normal.
Do yourself a favor and either check it out or get someone to check it for you.

 

A Mazda mechanic friend of mine told me that the LSD can also be adjusted to have more or less grab depending on your driving requirements. Maybe something to ask about if you are going to have it looked at by a professional.