You've selectively quoted the bits where I have been peeved with your 
lack of comprehension after several attempts.  All the bits where I have 
tried to explain to you the underlying physics seem to have passed right 
by you.

So I'll forget everything else and just try to explain the heart of it.

You misunderstand what angular momentum is.  Stop thinking that the 
angular momentum of a system is just the one part that is obviously 
spinning around.

The angular momentum of a system is the sum of the mass * velocity * 
radius of every single part of the system all taken relative to some 
reference point.

Mass and velocity should be clear.

Radius is the perpendicular distance from the reference point to the 
velocity vector.  Thus if the velocity vector passes exactly through the 
reference point the contribution to angular momentum is zero.  Any 
particle in the system that is moving and the velocity vector does not 
point exactly through the reference point contributes angular momentum 
to the total.  Now the contribution can be positive or negative 
depending on the sign of the velocity and which side of the reference 
point it falls on.

You can choose any reference point so long as it remains fixed.  It is 
often convenient to take the centre of mass of the system as the 
reference point when considering angular momentum.  Note that the centre 
of mass of a closed system is fixed within the frame of reference of 
that system.

Consider a closed system consisting of mass A that is not spinning and 
object B that is spinning.  Suppose A is the fixed frame of a satellite 
and B is the spinning wheel of a gyroscope.  All of the angular momentum 
of the system is contributed by the spinning mass B for now.  Let the 
total angular momentum be K.  K is currently not zero and lets say we 
choose our units and reference point etc so that it comes out as +100 
units.  The SI unit for angular momentum is Newton metre seconds (Nms) 
or Kgm^2s^-1.

Note that B possesses both angular momentum and kinetic energy.

Now apply a brake between A and B that brings them completely to rest 
with respect to each other.  The kinetic energy is converted heat.  As 
far as this closed system is considered there is no more usable kinetic 
energy available it has all been converted to heat.

Where is the angular momentum?  Well in applying a braking force to B, 
object A has experienced an equal and opposite force.  This causes it to 
start spinning.  In fact the whole system is now spinning compared to an 
external reference system.

The spin is in the same direction as the object B was spinning.  But A+B 
is not spinning at the same rate as B was alone.

How fast is it spinning and what is the new total angular momentum? 
Well it is spinning exactly fast enough so that calculating the mass * 
velocity * radius of every particle and adding them up comes to K as 
before.  Exactly.  There is no conversion of angular momentum to or from 
any form of energy.

Saying that K remains the same is not saying that there is any usable 
kinetic energy left in the system.  So forget the idea of perpetual motion.

The system had net angular momentum of K at the start, the end and at 
every intermediate point.

Considering it the opposite way around, if a satellite is spinning it 
can start to spin some part of itself in the opposite direction.  If 
this is done precisely enough then the satellite stops spinning but only 
so long as it keeps the gyroscope spinning at that speed and direction. 
  As friction slows down the gyroscope the satellite will start to spin 
again.  It can keep pumping in energy to maintain the rate of spin to 
keep itself pointed in one direction.  So energy keeps having to be 
introduced and it comes back out as heat due to friction.  This does not 
at any time alter the total angular momentum of the system.  AM has been 
transferred to the gyroscope and is stored there to come back out later.

If on the other hand thrusters are used to halt the rotation then the 
expelled particles carry away the angular momentum.  They are masses 
moving with velocity at a perpendicular distance to the reference point. 
    The particles in the thruster stream themselves don't need to be 
spinning to carry angular momentum relative to the reference point by 
the way.

Angular momentum is something different to energy and has its own 
accounting ledger.  The ledger always has to balance and you can't 
transfer amounts from the angular momentum ledger to or from the energy 
ledger.  Same with linear momentum.  It has its own ledger that always 
balances and likewise you can't transfer into or out of it from either 
energy or angular momentum.

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