Not a transmission guy but I'll take a shot at the second question.
Sounds like your alternator is being worked to death.
So let's start with the basics. Load test the battery - can be done for free at Auto Zone and all national chains. If it passes, great, but if it's marginal or wan't properly cured, then get an Optima. Try to get the one with both top and side mount terminals.
Always use a battery charger should the battery go dead. Using the car's alternator to charge it back up - strains the system.
Get yourself a decent Digital Volt - Ohm meter. Must have at least 10K "high impedance" look for a Blue point (Snap On Brand) or Fluke. on ebay, or Radio Shack has a decent one at about $70.
Start doing voltage drop tests at every connection. All kinds of sites on the 'web and you tube showing you how.
If money is too tight for buying a decent meter, then break out the elbow grease and sockets.
1. Clean the battery Terminals, check for corrosion where the cable instulation goes into the metal clamp.
2. Unbolt the Negative terminal, then start with the positive Battery cable Usually has two heavy gage wires, one to the Starter, & One to the Power Distribution Box. Make sure the connections are clean and tight. Don't trust your eyes, but disassemble and clean then down to 'bright' metal.
Remember, while the Neg battery cable is off, and separated from the battery, you won't get the sparks and fireworks should you accidentaly ground sumpn's.
Using a bit of steel wool, clean the area on the block where the Neg cable attaches. If you are in a high rust potential area, then get some 'conductive grease' from the hardware store. It's the stuff they use when assembling aluminum wires, this is the complete opposite of "Dielectric grease', but they both have the same intent: Keeping water and Oxygen out of the connection thus forming films that hinder the passage of electrons.
When reassembling the connection at the block, put the amount equal to two match heads on the bolt's threads. After you have cleaned them off course. Make sure you use a lock washer.
Now look for the straps that run from body to frame, or directly to the Battery's Neg terminal.
Again, you want everything clean and tight. Be careful as there is a lot of aluminum used in car these days, that stuff will strip out faster than snot ;^)
On certain GM products, look for a braided metal strap that runs from the back of the engine block to the firewall. If it's discolored in anyway, then the forgoing suggestions are mandatory.
For the record: I own no stock in any products I mentioned/will mention:
Hit everything with a blast of "PB Blaster" before beginning work or disassembly.
Don't buy it from Wal-Mart, it's the same price every where.
Use a product called "Liquid Tape" for any bare spots on your wiring if YOU CANNOT cover it with those shrink tubes. (They look like a straw, you slide a couple of inches of it over the bare spot - apply heat till it shrinks and make a tight fit.
Pretend you're a cowboy looking for stray cows but their really electrons which have met resistance, hence go looking for another path back to the Negative battery Post.
They will do a bunch of harm - if they don't get back the WAY the vehicle's designers and engineers intended.
If you have a CB radio installed or any serious add on after market stuff?
Do what the Audiophiles do:
Double up on the grounding system. They call it the Holy Three: Battery to Starter Motor: Nice big fat cable; Don't be fooled by thick insulation, talk to your parts counter guy. Just like a shotgun - lower number means more effect.
Next: Frame to Battery - again, get as big as you can for that.
Followed by: Body to battery. Keep in mind "Your car's entire body is an extension of the Battery's Negative post. Remember the battery with the top and side mounting posts I mentioned? Run these extra cables to the unused top or side post.
A couple of Warnings - Some GM products have modules which will go to sleep permanently requiring $ hundreds of dollars in replacement cost + a trip to the dealer for re flash of the car's computer, if they lose power .
Know your car.
Second:
When you are around batteries, wear approved safety goggles, when SHTF, you want your eyes to be protected. Have a hose nearby for flushing - just because S-H's.
You will need a decent Voltage meter to test for leaking AC voltage from the Alternator, this happens when the Diodes go bad.
IN MY Experience: Cars that eat up alternators will blow a head gasket 10x's faster than 'normal' cars. I suspect this is due to corrosion and electrolysis. So change your anti-freeze - on this car at every 20k miles. Check for a goo that looks like peanut butter with rust in it in the expansion tank. This is the by- product of the additives in the anti-freeze doing their job.
Just because the Anti-freeze is pretty bright green doesn't mean it fit for use.
That can be done for less than two movie tickets and a bucket of popcorn and sodas.
And No the stuff isn't end of the world toxic, in fact it's biodegradable. Careful using it, there's reports that dogs like to drink it and it is toxic.
HTH