I heard that a based unit cannot attack an opponent's based unit. However, I can seem to find this in my rulebook. Is this a real rule? If so, which rulebook/errata established that rule? If not, will the opponent unit get both the called shot and the friendly fire modifiers if I shoot at it?

It is in the rulebook, though it is worded very strangely:

You should lump all your questions into a single thread so we don't have to dig through the 15 different threads written by you every day.

The only loop hole I have found about the fire modifiers is if you are being based, and you want to shoot at an enemy mech you can ignore the modifer for called shot if you can target two things and are able to have range on the basing unit.

Example: You have a Clan Wolf Vulture. It is being based by an CNC ATV Scout. You want to shoot at an SC Uller. You declare a ranged combat with two targets, one the ATV and the other the Uller. The unit basing you is no longer making it difficult for you to shoot at someone else since you are also shooting him too.

To add on to Norcross's post: If you target multiple units, you're not required to damage all of them. This means that when you target the ATV and the Uller, you can choose to do 0 damage to the ATV and put all your damage on the Uller. Also, if your Vulture is based by multiple units and you want to attack the Uller, you only need to target one of the basers to get rid of the called shot modifier (you can find this on page 24 of the ROW under Ranged Combat Against Multiple Targets).

I'm fairly sure that you can't fire out of base contact into base contact even if you are targetting one or more of the figures in base contact with you. The problem is that even though you are avoiding the "called-shot modifier" the rules still say clearly that you can't shoot out of base into base. The fact that you are avioding the modifier does not change the situation.

you can not fire when both called shot and friendly fire are established. in other words if your mech is based by an enemy unit, you can not target another unit who is based by a friendly.