Is the resistance in a DC Motor constant if the voltage is?
The resistance in a DC motor is determined by the motor coil winding length and wire gauge. This doesn't change other than a small increase in resistance as the motor temperature increases.


  Is resistance in a DC motor constant if both the load and voltage are constant as well?


The resistance will not change (other than as noted above). For constant voltage current will increase with load.

Current and back-emf

Some basic DC motor concepts: (Try and grasp each one before going on to the next.)


DC motors spin when a voltage is applied to the terminals. 
When unloaded the motor speed will be proportional to the voltage applied.
If you disconnect the motor (from the supply) and spin it then voltage appears at the terminals. It is acting as a dynamo. 
The voltage that is generated in dynamo mode is proportional to speed. We call this "back electro-motive force" or "back-emf" for short.








simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Ideal motor and it's series resistance with power circuit.

Let's have a very simplified look at what happens when the motor is switched on.


The motor is at rest. Back-emf will be zero.
SW1 is pressed. A current will flow into the motor and we can determine this by Ohm's Law, \$I = \frac {V_{BAT}}{R} \$.
Once the motor starts to turn it will generate a back-emf. The "back" refers to the fact that it opposes the driving voltage. This means we have to modify our current calculation to \$ I = \frac {V_{BAT}-V_{BEMF}}{R} \$.
The faster the motor spins the higher will be the back-emf and the current will reduce. This is why DC motors take highest current at start or if stalled.
The motor will settle down at whatever speed allows enough current to flow to overcome losses in heat and friction.
If you increase the load then the "losses" increase. (You're doing work, remember, and you don't get it for free.) This will cause the motor to slow down. This in turn will reduce the back-emf so more current will flow.