Flash Drive or External Hard Drive

"Which one is faster at reading or writing?

This is the big question. Right now, operating systems are optimized for hard drives, and so speed gains using a SSD (solid state drive) (depending on the quality) are still relatively small over a standard hard drive. However, I have seen some very impressive read performance from high quality SSDs, even without optimizations. So SSDs have great potential, but HDs are still very good.

Which one tends to fail more often?

Again, depends on the quality. Currently, Solid State Drives have a Limit on write (erase) cycles. Wikipedia has a detailed breakdown of the advantages and disadvantages

Can an external hard drive work on windows and linux at the same time?

If you mean simultaneous access by a windows and linux machine, then what you're really looking for is a network storage device.
If you mean using an external hard drive on windows and then taking it over to a Linux machine. This depends largely on what file system is on the hard drive. I believe NTFS can be read and written to in Linux (any distro with NTFS-3g installed) as well as Windows.

How much impact can a hard drive sustain before it loses data?

Hard drives all have a measured 'Shock Tolerance'. Which varies from model to model. Usually higher quality ones have more Shock Tolerance. This is part of the technical specifications that you should be able to find for any drive model.

Do either of them suffer from overheating?

This is very rare for standard HDs and SSDs. However they do have measured temperature tolerance as part of their technical specification.

Do all external hard drives require 2 usb ports to operate?

No. The two USB ports are only used in cases where one USB port is not producing enough power for the drive.

Can the usb handle running programs, or will this degrade the life span further?

While it can, it is not as fast as using an internal drive. Most modern hard drives are designed for this. However much writing to an SSD will use up more writing cycles.

Is writing the only operation that degrades the life span or does reading also affect the life span?

reading does not effect the life span of a SSD."