I'm not a mechanic. But I've had my share of vehicle problems.

The top of the battery appears that your electrolyte (the liquid in the battery) has boiled over to the top of the battery. My guess is that it was overfilled with water at some point.

Which means someone tried to save the battery.. yes.. you definitely need a new battery. To see if that solves a lot of your problems, you could borrow a battery from someone for 5 or 6 hours and see how the truck behaves.

Now I don't know the history of the truck.. but lucky for you, you don't have a "maintenance free" battery. Which means, you can actually pull those covers off on the battery and check your water levels.

Water in the battery will evaporate over time, and if low enough, they won't hold enough of a charge (or enough cranking amps). If the battery hasn't sulfated itself yet... (the lead has corrosion, which is like rust but they call it sulfated).... you can fill each of the holes at the top of the battery with distilled water so when it charges... it will hold more amps and more of a full charge.

A maintenance-free battery is basically sealed, to prevent water evaporation (via liquid or gases). Your battery allows refilling of the cells. So make sure you pop those off and take a look. Be careful though... that's water-turned-acid in there, and you don't want your naked eyeball looking down into there. (I recommend plastic safety glasses you can get at a dollar store)... fill each of the holes / cells about 1" down from the top. Enough that you can see the water level. So it has room to gas off or bubble still...

Summertime and heat is the worst time for batteries to run dry and thats when you lose your cranking amps and charge ability.

What you really need in your community is to drive around and see someone in their driveway that looks like he's working on cars all the time. Go talk to them. He's probably an uncertified backyard mechanic. There is one guy like that every few blocks away. These guys love cars and trucks and being able to fix them. Yes... they are the case of beer people.

Also, find your nearest scrap yard or junk yard. Often crashed vehicles will have good batteries pulled out of them for 35% the cost of a new battery and they'll even guarantee it for 90 days... which is all you need to get going.

Your repair problems.. if they cost $600 could be dealt with for $50 to $100 if you found a backyard mechanic and scrap yard / junk yard in driving distance. I use youtube to figure out a lot of my own problems -- they can be a great help as well.

I read on @virtualgrowth's blog, there is the matter of inspection violations and other things. But hopefully some of the stuff I took the time to write you could help you think outside the box on how you can fix your battery issues for cheap... even if it is replaced with another used battery.

Careful... running the truck on the alternator alone, puts extra wear and tear on the brushes in the alternator and related wiring, etc. So if you run with a dead battery all the time, you start to overheat and overwork the alternator too.. soon you will be in the market for a new alternator. When your battery craps out, it is time to address the battery issue and not just keep jump starting it.

Let me know if any of this makes sense or helps in some way.