1) I am not familiar with this machine but I am guessing it has something to do with a rule on the playfield regarding scoring difficulty and or possibly replay.

2) The bank reset coil should trace back to a fuse on the playfield locally. Maybe check to see if that’s blown.

1. Spelling Teachers Pet on the backglass advances the drop target value from 50 to 100, 300, 400, 500 and finally to special. The advance goes in order as you spell. The when you hit the "T" target you get both Ts. When you hit the E target you get all three E's. So the last two letters needed to light special are actually S and P. S and P are in the outlanes, so you essentially can't complete the Special sequence after ball three without losing the last two balls and the game at the same time. So you will notice that there are switches behind the drop targets and inserts with S and P in front of them? On conservative: knocking down the drop targets and then hitting the switches behind them will allow you to collect S and P on the 5th ball only. On Liberal you can collect S and P on ball 4 or 5. So this gives you a way to complete the sequence after ball 3 without losing the game at the same time.

2. If the drop targets go up during the game when you hit the "up" switch, then the problem is most likely a switch on the game reset relay being open when it should be closed.

This is a great game. I have mine modified to continue giving 500 points after lighting special instead of giving zero points and only the special for each drop target. This way the game is better suited to play for high score or to use in a tournament.

https://pinside.com/pinball/forum/topic/teacher-s-pet-scoring-modification

I've had my Pet for 15 years, and never knew what that L/C setting did. I guess I'm lazy, but I never really dug into it to figure it out. Mine's been on Lib the whole time.
I'm going to play with that mod, though. I'd like to see points added plus hear the big knock!
Is there a purer feeling of joy than hearing a special knocker go off?
Jim