Cannot find any info on this unit, tried a couple sites.

Do I need to replace the control board and did replacing the bake element cause the other problem?
Sounds more like something in the back of the unit shorted out.
Replacing the element in and of itself should not have caused the problem.
Unplug the unit, take the back off and see if you can locate what shorted.
It does sound like the bake relay contacts on the control board have welded together.
This is hard to troubleshoot without a wiring diagram.
If the bake element is heating but not getting red hot it could be that it is shorted to the frame and getting 120 volts not the full 240 it normally receives.
Here is a good link with basic stove repair info
http://www.applianceaid.com/erange.html

Am I wasting money on an oven that's close to 20 years old?
I would unplug it, open it up and see if you can find any obvious faults.
If you can not find any then I would consider replacing the unit.
Cannot say what a new control board will cost you and there is a good possibility it may not even be available.

Took the back off. Someone spliced the wire that goes to one of the bake element terminals. It looks like maybe that shorted out--the inside of the oven back is black in that spot. I couldn't really figure out how to completely get to the electronics, but from what I can see, they all look fine--nothing looks black or burned. The insulation around the light bulb socket is also all black, but I think that happened a long time ago--haven't had light in there for years. What I don't understand is, if the wire to the bake element shorted, why is it still heating up when the oven is plugged in?

See Denman's post above:

Would there be some sort of circuit breaker or hi-limit switch that would protect the electronics from frying?

Your unit does have a hi-limit (Item 9 in Section 1) in the heater circuit.
Unfortunately electronic parts blow much faster than fuses do so it does not give any real protection.
The high limit is usually mounted on the outside of the top or back of the oven. It will kill power to both elements.

Repair the oven shorting problem, check that the hi-limit is 0 ohms and give it a try. I would also check the power to the unit L1 to Neutral and L2 to Neutral, both should be 120 volts. L1 side is required to power the board.
Who knows you may get lucky.

Note that the parts for this unit includes a wiring diagram.

Could you give me a guess as to what happened to my stove?
Without being there it would just be a shot in the dark.

How did the bake element get fused to the body of the oven?
Again I cannot say.
The bare wire section of the element cannot touch any part of the stove case/frame. But the insulated/calrod part of the element can, as you can see this is where it has the mounts to attach it to the rear oven wall.

As a way of explanation:
The unit gets power from two 120 volt supplies with a common Neutral which is connected to the frame of the oven..
One of these powers your light and control board as well as half the voltage for the element.
These supplies are 180 degrees out of phase. In other words when one is at positive 120 volts the other is at negative 120 volts. This gives you the 240 volts required by the heating element. Your current flows through the element from the -120 to the +120.
So if either end of the bare element touches the frame you have a dead short for one side of the electricity and it will arc/spark and normally trip the breaker.

I hope the above helps.