You will have to check the heater at the molex connector that connects the heater to the heater wiring harnes that goes to the control board.
Testing from the control board, the way it is set up, there is no way to tell if one element is burned out or not. Both coils in the element are 1000 watts each and are always on at the same time so checking from just the two heater wires at the control board will show the same ohm reading if both heating coils are good or either one is burned out, both conditions will give same ohm reading. Only condition that would show up is both coils burnt out.
If you check from 5 to 6 on the heater connector and get an open reading then one of the coils are bad. Checking from 1 to 5 and 1 to 6 on the heater molex plug will check each coil seperately, they should both be the same ohm reading.
To change the control panel you will have to remove the dryer. The main user interface/control board shouldn't be a problem for you to replace once the dryer is removed.
Check all the push buttons and make sure none are slightly jammed down which whould possibly be slightly pushing on the button inside the user interface/control board making the button have a hair trigger and activating the console. One of the actually micro buttons on the printed circuit board could have also collapsed causing the same type problem.
As with all brands of gas ranges with this burner sytle, the orifices start getting plugged over time from the impurities/oils in the gas sticking to the orifice opening causing the opening to get smaller over time - thus a smaller flame.
If this is on natural gas the easy way to clean the orifices without damaging/enlarging them is to unbend a small size paper clip and on one end make a loop in in so that you can't drop it down the oriface. Remove all the burner caps and heads, run the straightened paper clip in and out of the center of each oriface hole and reassemble. The burner flames will all be like new again.
If you range is on LP gas the right/rear burner is sometimes a small 5,000 BTU burner and the small size paper clip won't fit down the oriface.
And, as far as location, on yours the thermistor is included as part of the heater. To replace it you would have to replace the complete heater assembly Part ref# K320 in the 'Drum And Tub Parts' section.
The parts breakdown for you model shows the thermistor next to the heater but with a note for only a specific model number, (not yours), and the ref# K310 not circled which is also not listed in the parts list.
I don't do much LG work but from what I've seen with this problem in these forums I would be looking more towards the two solutions above my post not the actual thermistor but connections between the control and the thermistor connection.
Put the spanner nut back on but but only level with the top of the drive block, not tight.
Take your spanner wrench and place the end of one handle on the top of the drive block and spanner nut and smack it really good with a 2 lb. sledge hammer on the other end of the handle, go all the way around, then try the standard push down really hard and then jerk upwards on front edge of spin basket.
Usually just the strong push down at the front edge of spin basket and jerk back up will remove them but if that doesn't work I've never had the above method fail - I've had one or two that I was never able to get loose before I heard of the above method.
Also, make sure you haven't mistakenly swaped the L/F and R/F burner caps. If you have, the big R/F burner cap may sit and seal on the smaller L/F but the L/F cap if put on the R/F burner would be too small and not sit down correctly sealing the burner head to the cap and would let flame come out all the way around the burner instead of just out the correct burner slots in the burner head.
From what you have described I can tell you it's going to be the touchpad/console that you need to replace.
The keypad matrixes never go 1-2, 2-3, 3-4 or anything like that for each specific key, one group of keys will all be like, 1-2, 1-3, 1-4, then another group will be 5-1, 5-2, 5-3, 5-4. There's really know way to know what two traces go to any specific key without a techsheet that spells it out.
There are small diodes in the console keypad traces that keep the same keys that are all on one trace from feeding back to each other and one line trace can be bad which would cause multiple keys to be bad.
The test Chat_in_FL described above is sometimes the only way to know for sure if you want to prove it to yourself before purchasing a new control console. Your Kenmore is built by Whirlpool and most of the Whirlpool built dishwashers of your vintage and sytle the above test will work on. They don't always have a keypad matrix layout on the techsheets to even try to test with a meter for each individual key function.
Refrig and freezer sections are completely independent of each other. If the noise was still there when you had the refrig door open, (refrig fan would be off with door open), but stopped when you opened the freezer then the problem is with the freezer fan not the refrig fan.
About the only thing that could be dropping gray dust is one of the counter weights has broken and is crumbling apart. The wobble and noise would be expected if one of the counter weights has broken and is flopping loose on the tub.