I think I am probably confusing more than I am helping, but maybe this will be instructive and give you something to think about next time you try and rush a cut 
What’s important is not what your wall is acting like (it is a voltage source, though) but what the circuit controlling the motor is acting like… which is also (almost certainly) a voltage source. Your main question is: “What I want to know is if the speed control lowers the torque as well as the speed.” It’s not clear to me what you’re asking, but yes, there is a relationship between speed and torque for some motor. It sounds like you’re saying if I decrease the speed I decrease the torque, but that’s not true. It’s the other way. And it also works both ways. Let me try to explain.
A motor is only so strong. It can move light things quickly or heavy things slowly. Furthermore, it can only do this continuously in a certain range of operation. Say you are trying to lathe some light material. You set the speed control to spin at some speed. There is almost no mechanical load on the motor (we said our part was light), so it spins close to the ideal speed (speed constant * voltage). Now you dial your tool in and start cutting the material. This makes it harder for the motor to turn the piece. What has to happen? You’ve just changed the operating point. Something has to give.
In the equations page of my link, here’s the important part: v[sub]m[/sub] = iR + v[sub]emf[/sub]. (The middle term only matters as we move from one steady state to another. Let’s just examine the end result of these moves and ignore this term.) The v[sub]m[/sub] is what’s coming out of our controller. It’s fixed. The v[sub]emf[/sub] is sort of like the resistance of the motor to speeding… the faster it goes, the greater this term is. But we just admit we’ve slowed it down. So if v[sub]emf[/sub] decreased, and v[sub]m[/sub] stayed the same, then the iR term must have increased. Meaning: more current. (R is not a variable.) Our torque went up! But this is just what we required by loading it with our cutting tool.
Now, consider this: your v[sub]m[/sub] is changing because you’re changing the speed. We know that as the speed increases, v[sub]emf[/sub] also increases by some proportion (our speed constant). This means the iR term has to decrease. But this is related to the torque. So our torque went down!
This is what the graph on my link shows. For some fixed voltage, we can load our motor at the cost of the speed at which it will turn for a given input voltage. But you can look at it another way, and that’s that we can change the speed at which we turn provided that we don’t need a specific torque. If we need a specific torque, then we can’t control the speed.
The ideal speed controller would allow us to monitor the instantaneous speed of the motor and could deliver infinite current. So it would set up an operating point at some voltage (corresponding to our desired speed) and see how fast the motor is spinning. If it is too slow, it increases the voltage until the desired speed is reached. Of course our motor would then have to be able to move any load. Practically, of course, this cannot happen, and this is the limiting cases of our curves: a motor can only do so much before it overheats or whatever. The other thing a practical controller can do is cheat by not giving great speed consistency when loaded. It will drive the motor with no load at some point, and monitor the current going out. Increases in current (as we learned above: someone applied a load) will cause some feedback and boost the motor voltage up to try to compensate for this. When the absolute speed is not super critical, this is fine. (I have a fine application note from Burr-Brown on this called “Control a DC Motor without Tachometer Feedback” by Bruce Trump.) Even if the motor is way overpowered for our application, our controller has its own limitations relative to the amount of power it can deliver (power = voltage * amperage).
So there are real limitations all around. Real motors with real controllers:
- Control speed at the cost of fudging torque
- Can try to compensate, within reason, for change in torque by monitoring tricks