Most efficient way to power both 12V and 5V off a battery
First of all, you don't want to use a car battery, since they are designed to deliver brief high-current bursts ("cranking amps") and then be pretty much constantly charged by an alternator.  Instead, you want to use a deep-cycle marine battery, which are designed to be charged infrequently and then discharged down to 25% or so of their capacity.

EDIT: I modified the calculation so the batteries aren't discharged below 20% of their capacity.

You need a capacity of 7 * 12 * .5 * 20 = 840 AH.  The first battery I looked is a 6v one:

GPL-6CT:    300 AH    90 lbs  $380

It will deliver 8 amps for 2205 minutes, so it could deliver 10 amps for 29.4 hr; derating by 20% (since you don't want to completely drain it) gives 23.5 hr, a little over one quarter what is needed.  So you would need eight of these since they are 6v.  You would put two of the 6v batteries in series, and power a quarter of your display with that.

The second one is 12v:

GPL-4DL:   210 AH    124 lbs   $550

It will deliver 8 amps for 1375 minutes, so it could deliver 10 amps for 18.3 hr; derating by 20% (since you don't want to completely drain it) gives 14.6 hr, one sixth what is needed. So you would also need six of these.  However 20 installations doesn't divide by 6 evenly, so this is not a good option to consider.

It might seem to be tempting to run your 5v off the "bottom" 6v battery in each string using a LDO (low-drop out) regulator, but since the majority of your amps goes to the 5v rail, this won't make good use of the "top" battery.  So use a high-efficiency switching regulator (like the PYB30-Q24-S5-U) to drop the 12v down to 5v @ 6A.  You would use four of these, one for each set of 6v series batteries.

The above calculations and recommendations assume you will not be charging the batteries at all during the seven days.  If you do have access to a charger, then the AH requirements are cut by a factor of seven: 12 * .5 * 20 = 120 AH.  This can easily be handled by this 12v battery:

GPL-4DL      210 AH       124 lbs          $550    

It will deliver 8 amps for 1375 minutes, so it could deliver 10 amps for 18.3 hr; derating by 20% (since you don't want to completely drain it) gives 14.66 hr -- 
so you would only need one of these assuming you charged it each day.  (I though of another option where you would use, for example, 10 smaller batteries, one for every two installations, but then you would have to deal with 10 chargers which is not a good idea.)