That is the piece you need for replacing the part you broke, nice job finding it, Amazon is one of the cheaper places I usually look, there or RockAuto, if all else fails it is E-Bay.
I tried to find a good how to on here to help with replacing the PCV Valve, but all I could find if one for supercharged 3800 II'*.
I was able to find a site with a decent how to that should dhow you how it goes.
What you said about pushing down on the terminal, and hearing something click, it sounds like it did have a problem with it not making a good connection, so I'd go ahead and remove the battery, take it inside and give it 6-8 hours on your charger, of course this won't tell you if it has any dead cells or whatnot, it will charge it if the battery is still good, and as many times as I bet you have cranked it, I am betting it needs a good charging.
Put it on 6 AMP for 6-8 hours, or 2 AMP for 10-12 hours.
The charger won't show you the voltage of the battery though, you'd still need something like this to check the voltage it is holding when done.
I am not sure what device you are talking about that plugs into the cigg lighter, all you really need is what is below, and yes you can test the voltage the battery is holding while the terminals are connected.
Equus 3300 Hands Free Digital Multimeter, Electronic Digital Multimeter, Automotive Multimeter, Equus Multimeter
Also to really know if the batter is still good overall, you'd need to have it load tested, Advance Auto, or Walmart should be able to do this for you.
With what you describe about needing to pump the pedal to keep it running, it sounds like a fuel delivery problem, either there is not enough pressure due to something like a weak, or failing pump, or the MAF may be dirty, and or failing.
You did mention a mechanic seeing negative something in regards to the MAP and MAF, so maybe once you are sure the grounds are fine, and the batter is healthy, then cleaning the MAF, and if possible verifying the MAF is functioning correctly by testing it with a multimeter that can read in hertz, or a scan tool like the Actron one I have, it won;t read the hertz, but it will read the gp/* (grams per second) and that is enough to see if it rises and descends smoothly when you accelerate, and decelerate.
When you are done investigating the grounds, cleaning the burnt part on the post, and making sure the battery is fully charged I still suggest getting MAF Cleaner and cleaning the MAF'* thermistors off, the MAF is the main sensor responsible for the amount of fuel being delivered, and if it is really dirty, or failing, then it could cause it not to run unless you pump the pedal as you describe.
Your MAF is not exactly like mine, but to give an example of what I mean I'll tell you what happened to mine.
My MAF has a top that covers the thermistors, and it came off and was able to spin around in the throttle body'* bore for the MAF, so on occasion I'd have to pump the pedal to get it to start, then it would still run horrible for a while, until the tip quit spinning around and blocking the air from passing over the thermistors, eventually it got totally stuck blocking the airflow over the thermistors, and my car quit stating altogether, there were never any codes thrown, and it took me forever to figure out just what the problem was, it was then I learned just how important the MAF really was, and understood how a dirty, or failing MAF could cause problems with the car running right.
Even after figuring this out my car still ran rough, even after cleaning the MAF, the MAF was reading a tad low so it started to misfire at idle, the only way I fixed this was to buy a new AC Delco MAF, as soon as I did this the car idled perfectly, and did not randomly misfire anymore.
Of course this may be totally irrelevant to your problem, just trying to show the importance of the MAF functioning correctly.