Hey Steve,
I am by no means an expert...but here are some thoughts (in red)...
110V charger to 12v battery to 12v device (car radio)
No reason this won't work, as long as you are only using the car decks built in amplifier as adding any additional amplification will surely kill a small rechargeable 12V battery fast, and even then, I would only use a deck that was say, 15W x 4, staying away from 50W/75W x 4 decks.Such that all is wired inline so that when not plugged in the device just will run off the battery but when plugged in the battery will charge and if the device is on, will also run the device.
It looks like "generally" that's how these rechargeable batteries work. Whilst plugged in and charging, they are providing 12V pass through to device and once 120V power is cut to charger, device will draw power from battery. This is something you will want to double check when you order the battery and chargerCan I wire this in such a way that does not require a switch and so that when the battery is fully charged the charge circuit will stop charging?
I believe most chargers have over-charge circuitry built in, so you needn't switch anything on or off, it does it automatically...though, having a master "on/off" switch that completely cuts power to device would be ideal as to stop any parasitic loss (completely cutting power might erase saved station and pre-sets...might depend on the type of car deck you are using).Also, looking for off the shelf components that won't break the bank.
Lots of cheap examples on E-Bay and Aliexpress. Here is an example http://www.dx.com/p/1268-12v-6800mah-rechargeable-li-polymer-battery-blue-black-12v-227733?tc=CAD&gclid=CJng2OWF-swCFQ8yaQodUcoFvw#.V0ghNvNzbIU (though it does get horrible reviews, I am at work, so this is only example I can find quickly). Looks like they are readily available and cheap. I am sure this is a "You get what you pay for." type of item.What size battery am I going to need to run a car radio for a at least 4-6 hours without charging? The radio in question has a 10amp fuse but is it really drawing 10amps while operating?
I doubt you will ever pull 10amps, unless you are running this thing full blast and/or running speakers with a different ohm rating other than that speced for the deck you are using (most consumer car decks use 4ohm speakers, just double check as using higher or lower ohm speakers can damage deck or provide poor sound quality). As for battery size, needless to say, getting the biggest size battery (most ah capacity) you can is going to provide the longest running time.Does that mean a 12v 10ah battery would only run the unit for 1 hour?
Looks like when you try to do exact calculations on battery capacity run times, there are many factors that come into play. There are more technical people on the forum that could answer that better than me, but if we are doing dirty math, if the device really only draws, say, 3amps, then maybe you could expect 3 to 4hrs of run time? Really depends on quality and rating of the rechargeable battery.This is just some thoughts for discussion. Do you have a design sketched out for this boom box?
Also, some might chime in that there are lighter, cheaper, less power drawing options that allow for Bluetooth and music streaming services, but I am just answering based on your exact questions (though, most car decks now have Bluetooth and the one in our camper, as cheap as it is, will mute music and answer phone calls if phone is paired...which is pretty cool...except for wearing the microphone on you all day...lol).
Let us know how you make out...