Should laptop have 3-prog power plug?
 

My old laptop came with a brand new power supply (hence it is not original power supply sold by the manufacturer). The power specifications listed on the bottom of the power supply appear to match the power specifications listed on the bottom of the laptop. I know nothing about electricity, but do know that desktop comptuers tend to have 3-prong plugs. Should the laptop have a ground plug too? This has only 2 prongs. (I am in US)

Not necessarily. Old power supplies may not have 3 prong power cords.
If the output of the power supply matches the input range of the laptop you should be ok. Post the numbers for better a opinion

The laptop power supply will be case with plastic, therefore not really needing an earth.
(It's double insulated.)

http://en.wikipedia.org/wiki/Double_insulated#Class_II

If the case of the power supply (like in a desktop PSU) was metal then it would have an earth cable, so that in the unlikely case of a cable breaking inside it, and the live comes into contact with the metal case, the current would flow safely to earth.

At least, that's what I learned.

The other thing to condsider is that the level of current & voltage reaching the laptop is relatively small, when compared to what's coming out of the cord attached to a desktop machine. There is virtually no need of ground-fault protection from electrical shock due to a short inside a laptop, whereas if a desktop machine happened to short out on the A/C current side of things, the hazard would be much greater.

The external power supply's case is plastic, but the laptop itself is entirely metal. I'm not sure if that is a point worth noteing or not, but I believe all-metal laptops are not so common these days so maybe that might make exception, I have no clue.

Bottom of laptop:
DC=19V
3.16A
On hole for power input: 70 Watts or higher

Bottom of power supply:
Input: 100-240V~1.2A MAX, 50-60Hz
Output: +19V - - - 3.15A
Output power: 60 W Max

That power supply is probably fine for your laptop. If it was provided by the manufacturer then it should be fine. If in doubt, contact the retailer or manufacturer and ask.

As for the earth, the laptop is not at high voltage so it wouldn't need to be earthed. The transformer probably has a plastic case, so also wouldn't need to be earthed.

--Ian