The squeak could be the alternator or the water pump, get yourself a mechanics stethoscope and you'll be able to pinpoint the noise.
Sounds fine running cold and hot from what I can tell from the video.
The starting problem needs to be addressed. I'd start by making sure the battery is up to snuff and has at least 700 to 800 CCA rating. Then voltage drop the positive and negative battery cables (youtube the procedures), this will reveal if you are losing voltage anywhere in the starting circuit.
