Congrats on the FREE car! And a strong running Corolla with no apparent problems should last you a few more years, so good deal!
I'm guessing that you have a 4A-FE engine in your car (1600cc, 4cylinder, fuel injected engine)
On that car, there are 2 parts that work differently when the engine is cold. There is the cold start injector. This injects additional fuel when the car makes a cold start and does so until the engine is warm enough.
But since you say you always let the car warm up before driving, I don't think this is your problem.
The other is the Lambda (or O2 sensor) which is mounted in the exhaust somewhere before the cat. This needs to warm up before it starts to work propperly.There might be a problem with this sensor, which wouldn't surprise me with that milage.
A leak from the rad is never normal. How is the collant leven in the overfill tank beside it on the passenger side? Is it full or does the level vary when you have driven the car?
Does the bleed nipple itself leak?
The bleed for the clutch is located on the slave cylinder which is mounted on the front of the gearbox, next to the clutch lever which sticks out of the gearbox bell housing. When you stand in front of the car and look at the gearbox, you should see it right in front of you.
It is a hydraulic system that isn't really adjustable. You can adjust the clutch pedal. Your cousin might be reffering to that, but other than that, it should work as is.
With the milage your car has, you might have some wear on your synchromesh rings in the gearbox. Do you experience the problem in every gear? Or just certain gears?
Hope this helps!