Look into a class "D" amp. It will do the job efficiently. Speakers don't do very well without speaker boxes, you may find managing size-efficient speakers will be even more of an issue than the battery.
Non-sealed (flooded) lead acid batteries only generate hydrogen when charging rapidly or overcharging. Realistically, it is a small amount and mainly presents a risk if a small enclosed box is used.
Charging is difficult. As noted, this may blow the cigarette lighter's 10 amp fuse. Also, an alternator usually regulates at a voltage good for floating a battery, but is lower than desired for bulk charging. It will take a lot of engine time, in the hours.
You can get a charger specifically designed for it. If you can only find a 120V charger, get a power inverter.
Some older alternators burn up under surprisingly little added load. Something to consider.
A solar panel would need to be large. The rated current is for the best of direct sunlight, which doesn't happen many hrs out of the day.
Given your description, I would recommend sealed lead acid gel batteries, there's a size of around 17 amp hrs used for UPS & other bulk storage and accounts for 80% of SLAs sold. You may need a couple of them, which has the benefit of being able to charge one while using another. 3 hrs of music is not an unreasonable expectation, but it depends on amp & speaker efficiency and just how much sound we're talking about. It may last 1 hr or 12 hrs.
This is apparently an obscure definition of "camping" which I am not familiar with. Scouting has changed... starting to sound more like a rave.