How gently could a comet/asteroid/meteorite “hit” Earth? Could an object from outer-space with the right velocity and orbit come into contact with the surface of our planet in a manner that wouldn't cause it to burn in our atmosphere?
 A: It is difficult to see how. Most comets and asteroids would encounter the Earth on a crossing orbit and the encounter velocity would be roughly the vector sum of the Earth's velocity around the Sun (of order 30 km/s) and the individual velocity of the rogue object.  The individual velocity will vary from $\pm 30$ km/s for objects at a similar distance to the Earth from the Sun, to comets falling in from the Oort cloud on parabolic orbits which have roughly the escape velocity of the solar system at the Earth's distance from the Sun - i.e. $\pm 42$ km/s. 
This however does not define the minimum velocity. Even if you were to arrange it so that the asteroid/comet was diverted by something else so that it "caught up" with the Earth from behind travelling at an initially low relative velocity, there is the additional influence of the Earth's own gravitational potential. This would accelerate the approaching object to something like the order of the escape velocity from the Earth's surface - about 11 km/s.
Thus the range of possible velocities as the object enters the Earth's atmosphere would be from 11 km/s (near-earth asteroid travelling in the same direction as Earth) to 30+42=72 km/s (head-on impact with a long-period comet).
This fairly authoritative site on calculating the effect of asteroid/comet impacts suggests a minimum impact velocity of 11 km/s, which is indeed the Earth's escape velocity.
These are the speeds at the top of the atmosphere. If the objects are smaller than 20-30m, then the atmosphere will basically take out most of that kinetic energy, so for small objects, a relatively slow impact at a terminal velocity of some hundreds of km/h is de riguer. But for anything larger than $\sim$50 m, it is basically the full 11 km/s or more. http://www.lsst.org/lsst/science/scientist_near_earth_objects_neoquant
A: I think you could possibly engineer such a collision between two bodies in horseshoe orbits.  The minimum mutual speed between the two bodies depends on their mass difference, with a limit that approaches zero speed if the bodies are the same mass. I haven't tried very hard to do this, but it's the only possible way around the escape-velocity argument put forward by John Rennie.
Two images lifted from Wikipedia:


The Lagrangian points are there to motivate an "effective potential" argument, which I'll let you go through on your own.  Here's another approach: suppose that there are two equal-mass Earths, both orbiting the Sun in zero-eccentricity circles at 1 AU with a period of one year. Gravity wants to pull the two Earths together, by slowing down the leading one and speeding up the trailing one. However the one that slows down no longer has enough kinetic for its circular orbit at 1 AU — it's now at the highest point of an elliptical orbit with semimajor axis less than 1 AU, and period less than one year. Likewise, the object that sped up is now at the lowest, fastest point of an elliptical orbit with period more than one year. They move apart! In a rotating reference frame which includes the central mass, the gravitational attraction between these two objects is effectively repulsive.
I'm not sure what constraints there are on the closest approach in a horseshoe orbit, which would determine whether a zero-velocity impact is possible for rocky bodies in our solar system. 
It's not possible for Earth and rocky bodies in our solar system, since Earth does not fill its Hill sphere.
I'm pretty confident that there's some combination of relative masses and densities where two objects orbiting a common center could have their surfaces touch at zero velocity and then be gravitationally induced to move apart. That would still be a destructive interaction, though.
A: The problem is that the Earth's surface has a large and negative potential energy. Suppose we start with our comet a long way from the Earth and moving only very slowly i.e. we give it the lowest energy possible. The gravitational potential energy at the Earth's surface is:
$$ U = -\frac{GMm}{r} $$
where $M$ is the mass of the Earth, $m$ is the mass of the comet and $r$ is the radius of the Earth. If we let the comet fall from it's initial position to the Earth's surface the potential energy has to turn into kinetic energy otherwise energy won't be conserved. So the kinetic energy of the comet will be given by:
$$ \tfrac{1}{2}mv^2 = \frac{GMm}{r} $$
Plugging in the values for $M$ and $r$ we find the velocity at the surface of the Earth is around 11 km/s or about 25,000 miles per hour. And that assumes the comet starts off stationary. It's more likely the comet will have a considerable initial velocity and will hit the surface even faster.
A: The Problem is the orbits.
The Earth is traveling in a nearly circular orbit about the Sun.
Comets and asteroids travel in highly elliptical orbits.
This means when their paths cross, the velocity vectors
are large, and they are not going in the same direction
as the Earth when they both try to occupy the same space
at the same time.
Very high velocities combined with in-compressibilities of
solids and liquids equals a vaporized comet or asteroid,
and a large impact crater with lots of ejecta sprayed
out radially ( conical ).
Only the very smallest of space rocks can slow down in earth's
atmosphere and land on the ground without making a dent.
A: A system with such massive masses is subject to non-trivial amounts of gravity pulls. The extreme theoretical and HIGHLY improbable case is for the comets path to be tangent to the event horizon of Earth and from that point on to be trapped into a geo-synchronous (with a slight offset) declining orbit around Earth eventually leading to a "touchdown".
The massive amount of variables needed to be fine tuned makes this quite the coincidence!
