A constant current means, for an ideal motor, a constant torque. This is approximately true for real motors. It doesn't matter what you attach to the motor, or how fast it's turning.
What you seem to be missing is Newton's second law of motion. It states that force is the product of mass and acceleration:
$$ F = ma $$
The constant current you supply to the motor is one force. The weight opposes that force. The difference is the net force, \$F\$ in this equation, and \$m\$ is the mass of the weight, plus the mass of the rotor and the string and everything else the motor must move.
You set current to be sent to the motor so that the torque applied is 10 in-lbs without any load.
Not possible. There is nothing for the motor to "torque against". This is the mechanical equivalent of trying to develop 10 volts across a dead short. The motor will rapidly spin at its maximum speed, and the back-EMF will rise to the driving voltage such that your driving electronics are unable to supply enough voltage above the back-EMF to make enough current to have that much torque.
Let's just say you determine how much current is required for 10 in-lbs of torque, and you drive your motor with a constant-current supply set to that.
What happens when the torque from the weight/load is 5 in-lbs?
Assuming that the rotor and the string are massless and frictionless, the weight will be accelerated upwards by the net 5 in-lbs of torque (motor's 10 in-lbs, less 5 in-lbs from the weight). The rate of the acceleration is determined by the mass of the weight and Newton's law above.
As the speed of the motor changes (the weight is accelerating), the back-EMF also changes. Your constant-current supply to the motor will have to apply an increasing voltage to maintain the same current. Electrical power thus goes up, as does mechanical power.
What happens when the torque from the weight/load is 10 in-lbs?
Motor torque balances weight torque. However fast the weight is moving (if at all), it keeps doing that. Newton's first law applies.
What happens when the torque from the weight/load is 15 in-lbs?
The weight will accelerate downward, overpowering the motor. However, it won't be a free-fall. The motor cancels some of the force of the weight, resulting in a slower acceleration downwards.
If the weight overpowers the motor, then eventually it can get the motor to run backwards, relative to the way it would run if there were no load. When this happens, the back-EMF now adds (instead of subtracts) from the voltage you apply to the motor. At some point, your controller, which is attempting to maintain a constant current, must apply a negative voltage to maintain that current. In other words, the back-EMF is sufficient to create the necessary torque on its own: your controller must oppose it.
This is perfectly symmetrical with the first case, where the motor was overpowering the weight. In that case, electrical and mechanical power went up (without bound, if you let them). In this case, electrical and mechanical power go down (negative, if you let them). Energy is conserved because you are changing the gravitational potential of the weight.
The need to resist the back-EMF usually means storing electrical energy in a capacitor or battery, or using it to heat a resistor. If you can't do this fast enough, then the motor will create more torque than your desired 10 in-lbs, and you have hit the limits of your "constant current" driver.
Further reading:
The basic principle is simple - torque is proportional to armature current * magnetic flux. In a permanent magnet or shunt wound motor you can assume that flux is constant, so torque is just proportional to current.
However this does not take into account internal friction, windage, and magnetic losses. When the motor is running free these losses cause it to draw a no-load current (Io). Subtracting Io from total current draw leaves you with the portion that produces output torque.
Power = rotational speed x torque. As more load is applied the motor draws more current, which increases torque. However as current flows through the windings their resistance causes the effective voltage to drop, so speed decreases. Below 50% rpm the power output will also decrease, reaching zero at stall.
In a series wound motor the situation is different, because flux is not constant. With no load a series wound motor will speed up until friction and windage losses match the internal torque supplied by Io. This rpm could be very high, perhaps even high enough to destroy the motor. When a load is applied the resulting torque is proportional to current squared, because both armature and field currents contribute to magnetic force.
Best Answer
The link in case 1 is pretty thorough - and includes the note that a true thermal model of a motor isn't that simple. There are three three losses in motors considered mechanical - iron losses (in the armature core), friction losses (bearings and brushes) and windage. These subtract from the electrical torque generated by the motor - to zero in the case of no load operation, obviously. Iron losses are fairy invariant with temperature, windage falls with temperature, but is probably negligible on a small motor - unless it has a shaft-mounted fan, and the bearing friction will also fall with temperature. Brushes are far from predictable, and their coefficient of friction can vary widely with temperature and surface condition of the commutator. Complicating all that is the temperature coefficient of the magnets. Most materials will reduce strength with temperature, and this affects the Kt value noticeably - so more current will be needed to generate the same loss torque. So will the no-load current rise or fall with temperature? That will all depend on the balance of the losses.