Pins are not switching
What line ending are you using in your IDE? Because if it is other than 'no line ending', you are also sending a new line, carriage return or both. Those control caracter get included in mystring.

Better to be safe and filter out control caracter. Try this:

String mystring = "";

void setup() {
  Serial.begin(9600);
}

void loop() {

  while (Serial.available()) {
    delay(10);
    char mychar = Serial.read();
    if (mychar == '\0')
      break;

    if (isAlpha(mychar)) {
      //  Doesn't include control caracters in mystring
      mystring += mychar;
    }
  }

  if (mystring != "") {
    Serial.print("***"); Serial.print(mystring); Serial.println("***");
    if (mystring == "F") {
      Serial.println("mystring is equal to 'F'");
    } else {
      Serial.println("mystring is not equal to 'F'");
    }
    mystring = "";
  }
}