If I choose an advanced class that uses a different weapon than the standard weapon you use now, will you still be able to use the weapons you like?

 

ex. I'm a Republic Trooper, who wants to become a Commando. Commando's use the Heavy Firearms, basic troopers use assault rifles. If I become a Commando, can I still use assault rifles, or am I limited to the Heavy Firearms?

as a commando you can still you rifles, but there are commando abilites that require the use of assult cannons to use them. So you will be limiting yourself by using rifles as a commando

Does the same rule apply with Lightsabers (double bladed, X2, single)?

You never lose the ability to wield something, so when you unlock double-bladed sabers for example you can still just wield one, if you wanted.