I was curious, why does a craft have to attain 7 mi/sec to escape the Earth's gravity? Could it go slow and powerful and still escape?

And could an object be slown way down upon re entry and not get hot?

There is a minimum velocity needed to escape the Earth's gravity. This velocity varies according to the distance between the object and the Earth. So, for example, it takes more energy to get from a launchpad on the Earth's surface to the escape velocity than it would to leave from the Space Station or another point in Earth orbit.

This also means there is a lower limit on the "slow" in "slow and powerful."

There are lots of ways to "slow down" on re-entry. But they all involve at least falling through the upper atmosphere at a pretty good clip. If you made a powered descent (thrusting against gravity) it would require so much energy you couldn't carry enough fuel to make it all the way down to an altitude where a parachute could take you the rest of the way without heating up the re-entering vehicle.

But does one need to go 25000 mph to escape the Earth gravity? What if there were a thousand mile high ladder, could one simply climb it until he escaped gravity?

The Space Elevator is one idea for getting payloads into orbit, and it's very like "climbing a ladder". The problem is getting the "space anchor" end of it into orbit and keeping it there.

Your ladder would need to be considerably longer: many communications satellites are 12,000 to 18,000 miles out and still orbiting. The Moon, of course, is even further away and still orbiting. Etc.

Theoretically, one can envision a "ladder" that is so high that eventually you could just "step off" in a direction perpendicular to it and achieve escape velocity (remember, the velocity needed to escape is related to the distance between you and the Earth -- the further away you are, the less velocity is needed).

Speaking of "long" like a ladder, this reminds me of another theoritical situation. If you had a string pulled taught from Earth to Jupiter and the person on Earth gave it a quick tug, how long would it take for the person on Jupiter to feel it? Would it be instantansous?

That would depend on the elasticity of the material in the string and on the strength of the "tug" ... the string itself has inertia (the molecules of the material in the string interact with each other in an elastic manner, so there is some minimum amount of stretch in the bonds between them which must be overcome before movement of one molecule is felt by the next ... and in a long chain of molecules there is an additive component to this effect.

To take this to silly extremes, imagine on string made of iron and another made of a stream of water ... even a slight tug on the iron string would eventually lead to motion at the opposite end, whereas the same tug on the string made of water might simply pull it apart.

This is a better question that it appears to be on the surface. Google "space elevator ribbon" or "carbon nanotube" and you'll see what I mean ...