Re: why does an increase in temperature increase the rate of osmosis

You've already got the reason, kinetic energy does increase with 
increasing temperature.  But what exactly does that mean?

Think of a sheet of paper.  Now poke very small holes in it.  Think of 
that as a somewhat permeable (semipermeable) membrane.  Now, for 
convenience and not to make a mess, hold the paper horizontal to the 
floor, just like you'd have it on a desk.  Now have someone sprinkle the 
paper with those little red or green sugar sprinkles that you put on 
cookies.  Maybe one or 2 will fall through one of the holes, but not 
many.  Now, SLOWLY, start moving the paper back and forth.  The sprinkles 
with start moving, but not very fast.  This will cause some of them 
to "find" a hole and fall through.  But it will be relatively slow.  Now 
speed up the movement of the paper, shake it back and forth like you're 
sieving flour.  The sprinkles will move faster and faster (increased 
kinetic energy from the increase in temperature).  The faster they move, 
the more area they cover in a given unit of time.  Therefore, over that 
period of time they are more likely to find a hole in the paper/membrane 
and fall through.  It's a simple probability idea.  The higher the 
temperature, the more likely any given molecule/sprinkle will find a 
hole.  At some point of course, the molecules that are diffusing through 
the membrane will be equal on both sides and the system has come to 
equilibrium.  BUT, remember that this doesn't mean that nothing is moving 
through the membrane, only the the amount moving in one direction is equal 
to the amount moving in the other direction.

Now this example is actually somewhat backward in that we're moving the 
paper/membrane rather than directly moving the molecules/sprinkles around 
in a solution, but it's exactly the same idea either way. Further, osmosis 
doesn't necessarily imply that a molecule is moving through an actual hole 
and most of time it doesn't since we're usually talking about water.  
Osmosis is simply a molecule, let's say water, leaving one solution, 
dissolving in another (the lipid of the membrane), moving around for a 
while in that solution and then leaving the lipid and redissolving in the 
solution on one of the other side of the membrane.  Once it's in the 
membrane, it has a roughly equal chance of exiting on either side.  
Osmosis is simply reflecting the idea that there are more molecules on one 
side than the other.  Therefore more molecules dissolve in the membrane 
from one side than dissolve from the other.  Since they have an equal 
chance of exiting either side, then numerically, more molecules from the 
high concentration side will end up on the low concentration side, until 
equilibrium is reached.

There is also an additional component, but it's more dependent on the 
specific molecule and the composition of the membrane.  As temperature 
increases, a membrane becomes more fluid, more loosely packed and the 
lipid molecules move around faster (like butter melting in the frying 
pan).  This has the effect of making the membrane more permeable.  For 
some molecules it also will make the molecule more soluble within the 
membrane.  This will also increase the rate of diffusion.  In effect, what 
this component does is alter the properties of the hole in the membrane if 
we use that type of explanation.  Alternatively, in addition to simply 
increased rate of movement of the molecule, the increased temperature 
makes it more soluble in the membrane so that once it finally swims around 
and bumps into the membrane, it has a greater chance of dissolving.

Hope that helps.