Can a majority of DC electronics run on high current (i.e. lamp, car) batteries directly?
Whether you can run it safely off batteries depends on the design of the unit.  


If the regulator is in the supply then you will need to add a regulator to your batteries.
If the regulator is in the unit you will be able to directly attach your batteries.


Most systems have the regulator in the unit, so it should be OK to just power it off battery.

Be warned though that "12V" lead acid car batteries are actually a higher voltage (around 13.5V) so a regulator should really be used with these for safety.

As for increasing current - yes, just put pairs of 6V batteries in parallel.

Connecting batteries in parallel increases the current.  Connecting them in series increases the voltage.

If you think of a 12V battery as two 6V batteries in series, then connect these "12v batteries" in parallel you will get 12V and higher current.

Electrically, a single 12V battery is the same as 2 6V batteries in series.  A battery is made up of cells.  Each cell can provide between 1.2V and 1.5V depending on chemistry.  A 6V battery is a battery of 4 cells (battery in the non-electrical sense - think artillery battery, a group of artillery guns) - 4 x 1.5 is 6V.  Put two 6V batteries together and you get a 8 cells in series - that's 8 x 1.5V = 12V.