Updating Data Input - Serial port

It bothers me that you cannot stop the Arduino from executing the program.

Does it bother you when a light bulb exhibits the same behavior? Turn it on, it does what it was designed (programmed) to do, until you turn it off. This seems like a desirable attribute to me. I mean, image having to periodically turn the light on again to make it put out light.

Once you've typed in a text, it translates and blinks, but you cannot make it stop

Because you have programmed it that way. You could just as easily have programmed it to send the message ONCE.

It would be cool if it would notice that there's new data available at all and then start again,

So, program it that way.

It would be even cooler if I could type in a certain command like 'Stop!" and make it stop executing and blinking so that you could start typing a new text aferwards!

So, program it that way.

I've read a bit about interrupts, but I don't know how to make an inerrupt through the Serial port

There is already an interrupt triggered when serial data arrives. You can't attach another one. And, not that way, even if you could.

The first argument to the attachInterrupt() function is the interrupt number. Serial is an instance of the HardwareSerial class, not an int containing an interrupt number.

String string;
String input;
String code="";
String morsecode="A. -,B- . . .,C- . - .,D- . .,E.,F. . - .,G- - .,H. . . .,I. .,J. - - -,K- . -,L. - . .,M- -,N- .,O- - -,P. - - .,Q- - . -,R. - .,S. . .,T-,U. . -,V. . . -,W. - -,X- . . -,Y- . - -,Z- - . .,a. -,b- . . .,c- . - .,d- . .,e.,f. . - .,g- - .,h. . . .,i. .,j. - - -,k- . -,l - . .,m- -,n- .,o- - -,p. - - .,q- - . -,r. - .,s. . .,t-,u. . -,v. . . -,w. - -,x- . . -,y- . - -,z- - . .,0- - - - -,1. - - - -,2. . - - -,3. . . - -,4. . . . -,5. . . . .,6- . . . .,7- - . . .,8- - - . .,:- - - . . .,;- . - . - .,?. . - - . .,_. . - - . -,(- . - - .,)- . - - . -,\'. - - - - .,=- . . . -,+. - . - .,/- . . - .,@. - - . - .";
String incoming;
String incomingcheck;

Loose the String class. Period.

  if (Serial.available()>0)
  {
    input=Serial.readString();
    length=input.length();

You really need to define some end-of-message marker (the carriage return and/or line feed that the Serial Monitor may add would be fine), and read and store data in a char array until that end-of-message marker arrives. Reading an arbitrary string from serial is not a good idea. The function waits up to one second for more serial data, even when there is no more coming. You could know to stop waiting long before that.

    Serial.flush();

Why is important to block, waiting for debug data to be shifted out?