How to make timed loops after switch / case?
Create a new variable to hold the state of whether or not you are in autoClose mode:

uint8_t inAutoClose=0;


Then, add a line in your switch statement to set that flag when you enter that state, but not call the function if you were already in that state:

case 10:
  //automatic close
  Serial.println("A Close1"); 
  digitalWrite(gear1ClosePin, OFF);
  digitalWrite(gear1OpenPin, OFF);
  if (!inAutoClose) {
    inAutoClose = 1;
    t.enable(autoClose);
    autoClose();
  }
  break;


Add a line in your other switch branches to clear that flag:

case 1:
  //manually close
  inAutoClose = 0;
  [...]


If you want autoClose() to be called every 10 seconds repeatedly as the "idle" state, then you need to clear inAutoClose (set it to zero) at the end of every 10 second period. How and where you do that depends on other factors you haven't shown, like what autoClose() does, or what the timer you enable does.

Side note: avoid naming your timer name (autoClose) and your function name autoClose() the same! Does your code compile that way? It ought to throw an error.

ANSWER 2 (based on comments)

Your auto-close state is described as


  Turns on for 10 seconds, after that off for 10 seconds.


This can be accomplished with a similar structure to what you have now, with a few changes. (These are my suggestions, but by no means the best or worst way.)

Edit your open and close functions:

void Open1() {
  digitalWrite(gear1OpenPin, ON);
  if (isAutoOpen) {
    t.setTimeout(10000UL, Close1); // Sets a timer to call closer in 10s
   }
}

void close1() {
  digitalWrite(gearClosePin, ON);
  if (isAutoOpen) {
    t.setTimeout(10000UL, Open1); // Sets a timer to call opener in 10s
  }
}


When you call either Open1() or Close1(), if your auto-open flag is set, then 10s later, the opposite function is called, and this keeps chaining together, every 10s.

Whenever it's time to stop this, you clear the isAutoOpen flag and disable the timers. You may wish to keep track of the timer ID that is created with the setTimeout() function, so you can disable it at any time to avoid the system going through the cycle one more time after you clear the flag.