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The water pump is probably bad and needs replaced. It could just be a faulty gasket but, with the age of the vechicle I would change the pump as well as you will have to remove to replace gasket anyway.
Sounds like the linkage has broken or come loose. If the driver side is working, there is only one motor that moves both wiper arms. Need to find the loose or broken spot in the linkage.
You can find out. Take vehicle to an Autozone and get a free read of the check engine codes and post result on here to further assist you.
Check oil level. May just be a little low.
Is it very loud when you are running, like an exhaust breakage before the muffler loud?
Use that warranty, and take to the dealer. Be as descriptive as possible to the mechanic.
Definetly sounds like an alternator issue. Did you 'jumpstart' the vehicle before you replaced the battery?
I will assume you mean recharge the AC system. From the air compressor you will see two metal lines coming out. A thinner one and a wider one. The wider one is the low-side (lower pressure side) Follow the wider one till you find the schrader valve.
What is the question?
Then there is a leak in the system. The leak must be repaired and sealed or the AC will never hold a charge. Take to a shop and tell them you need the AC system checked with a dye test to locate the leaks.