Will a 12 volt battery get charged if the booster is set at 6 volts?

No.

It takes approximately 14.4 volts to properly charge a 12 volt automotive battery.

It takes approximately 7.2 volts to properly charge a 6 volt automotive battery.

You can't even charge a 12 volt battery with 12 volts.

A 6 volt charger would give approximately 7.2 volts, which is not nearly enough to charge a 12 volt automotive battery.

I think that 12.9 volts might be enough to start charging a 12 volt automotive battery.

No it will not.

Nope. Why would you even try that? 12 volts is 12 volts. Clean the battery posts first and if the battery has a removable cap to add distilled water top the fluid level up too.

No, the fundamental rule is that the voltage applied must be higher to get any current to flow. A 6V trickle charger has voltage just above the float charge voltage of the battery its for, in this case about 7 volts. You can't charge a 12 volt battery with a 6 volt charger. it will be never charge

Depends, some old cheap chargers regulate current but not voltage. Big problem if you walk away and leave the battery on the charger, it will boil dry. An old 5 amp charger for example may have an open circuit output of 40 volts, the moment you connect it to a battery it regulates charge current down to 5 amps and if you measured it's output voltage it would be something slightly above the battery. As the battery comes up to it's fully charged state the charger doesn't know, it just keeps raising it's voltage to maintain 5 amps. These old beasts usually had a timer switch on them so you could set charge time.

Well I've had it happen so wonder how or why.

Nope. The way to charge any battery is to have a HIGHER voltage than the "ending" result. Most fully charged 12v batteries, for instance, when measured, are 13 point something volts if you actually MEASURE the voltage. Even 1.5v batteries when new...are 1.6 or so volts.