your comment:
It almost feels like a dead battery, except that I had put in a brand new one (and plugs) less than a month ago when the coil first started acting up. I'm not sure how to check for electrical wiring issues, since I'm finding the wiring schematic kind of hard to trace as it doesn't pinpoint the physical location of each point on the circuit.
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here are some possibilities:
on most craftsman mowers, there are two safety switches that must be closed in order for the start circuit to operate - (1) clutch/brake pedal depressed and (2) blade engage lever off.

next the battery cables must be tight with no corrosion. sometimes the wires appear to be secure, but there is no continuity. tighten up all connections.

finally, the ground for the starter relay is often located under the right rear fender - just over the right rear wheel of the tractor. The ground is bolted to a painted surface, and after a few years, continuity is lost. locate this connection and tighten.

an ohm meter is useful to troubleshoot. Follow safe practices when using wrenches around batteries.

The 42e707 was one of the better engines for most homeowners.

mownie, thx for the prompt and detailed response. Just so I get this straight, are you thinking that the gunk build-up in the carb jet is causing the irregular (up & down) engine rev that I described as the initial symptom ? Right now, I can't even get it to turn over when I switch on the ignition.

Thanks again for your help.

Yes, I am referring to the surging, and when you said the whole thing sort of died on you........well, I have seen that happen too in the case of gunk settling in the carb and blocking the entrance to the main jet.
And after that I sort of lost my thought train because a guest showed up. So I was trying to hurry and get my post done.
Now that I can sort of think again.......
Did you disengage the mower deck after it died.....before you tried to restart the engine?
The manually engaged deck has a switch that inhibits the starter solenoid from activating if the deck is in the engaged position.
bluemower covered the electrical highlights pretty well.

bluemower, thx I already know about those 2 safety switches. I think there's also one under the seat, but not sure if that only cuts in when the blades are engaged ? since I remember I was able to get off the seat to retrieve my hat w/o engine shutting off. I also cleaned the battery terminals but didn't use baking soda as there wasn't that much build-up. Also made sure the fuse was secure, and will try and locate the starter relay ground wire you mentioned.

Thx again for your help, and any other tips would be much appreciated.

I've been trying to narrow down the problem, but haven't found anything positive so far. Based on the voltage drop, the wiring for the following parts look fine:
1. Battery and battery terminal connections.
2. Fuse (both voltage and continuity)
3. Ignition switch (live and ground).
5. Battery end of solenoid.
4. Cable from solenoid to starter.

If the solenoid checks out then it must be related to the starter or one of the switches, but it's hard to access from where it's tucked under the battery, and I haven't found the other end that connects to the interlock switch to confirm if the solenoid itself is bad. There is also no voltage at the ammeter, but I'm not sure if that needs the ignition to be on ?

ALL current traveling the circuit between the battery side of the starter solenoid and the ignition key switch MUST pass through the fuse and the ammeter. If you can detect 12 volts at the fuse, and 12 volts at the "B" terminal on the ignition key switch...........then you must also be able to detect 12 volts at the ammeter. If you are checking for voltage at the ammeter by connecting your volt meter or a test light to both posts of the ammeter at the same time, you will not detect voltage.
To test for voltage at the ammeter, one lead from the volt meter (or test light) must be connected to a ground and the other lead contacting to a post on the ammeter.
Testing should show that voltage is present on each post of the ammeter. The ignition key switch does not have to be on to test for voltage at the ammeter.

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Here is a comprehensive test procedure for testing the starter control circuit of your tractor. Refer to the image below the procedure for a graphic referrence to the steps.

For this, a 12 volt test light will probably be easier to use than a volt meter. So that's how I will describe the tests. The battery must be fully charged for these tests to be accurate and you must make sure there are no blown fuses. Clip the ground of the test light lead to a known good ground, negative post of battery is ok. Confirm the test light works by touching the probe of the light to the battery positive post. If light works, proceed to tests. You might want/need a helper to turn the key switch if you have short arms.
Lock the brake pedal and make sure the PTO switch or, or other engaging method is not engaged.
Step 1 Find the "S" terminal on the back of the key switch (a white wire). Insert the probe of the test light inside the wire connector where the white wire enters the connector, make certain the probe is deep enough to touch the metal terminal. Turn the key to the START position. The test light should burn. If test light does not burn, replace the key switch.
If light burns, go to step 2.
Step 2 Touch the probe of the test light to the small white wire on the starter solenoid, turn the key switch to the START position. If the test light burns, the entire control circuit for the starter solenoid is good. If this is the case, move the test light probe over to the large post on the solenoid that has the cable going to the starter. Turn the key switch to START position. If the test light burns now, the starter is faulty, or the cable from solenoid to starter is defective. If the test light does not burn now, the solenoid is faulty.
If the first test of the white wire on the solenoid did not cause the test light to burn, go to step 3.
Step 3 Find the brake switch and locate the 2 white wires on that switch (there might be 4 wires on the switch, only check the white wires). Turn the key switch to START and hold it in that position. Touch the probe to each white wire. If the test light burns when touched to both wires (one wire at a time of course), the clutch/brake switch is good. This would mean there is an open in the white wire circuit between the brake switch and the PTO switch (the switch on the deck, because you have power on the white wire leaving the brake switch but no power on the same white wire arriving at the solenoid).
If you have power showing on just one white wire at the brake switch (with pedal locked down), make sure that the switch is adjusted properly. If adjusting the mounting of the switch does not give power to both wires, replace the brake switch.
Now, if you have power at both of the white wires on the brake switch , move on to the PTO switch in step 4.

Step 4 Find the white wires on the PTO switch and place the probe on the white wires (one at a time). Turn the key to START. The test light should burn on both white wires. If only one white wire has no power on it when in START position, replace the PTO switch. If you have power on both wires here, but not at the starter solenoid, there is a break in the wire between the PTO switch and the starter solenoid.

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If it is a dead battery, I heard that there is a fuel switch that will starve the engine if the battery isn't functional (solinoid) perhaps you can put the battery on a trickle charge, the manual even says that if you store the tractor for long periods the battery should be on one of those low amp charges, like the one I bought from harbor frieght for $15, meant for boats but works great to keep the battery on the tractor healthy.