Help with a up and down counter

Would a 74AC08 quad AND gate be suitable? And what would be a good choice for the FET's?

74AC08 would work.

I was looking for suitable FETs and found these at DigiKey.
 
Have you measured the current drawn by the three solenoids with 13V applied? All you need to do that is a fully-charged 12V battery, two clip leads, and a Multimeter capable of reading 10A full scale.
 
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