Thanks for the answer. You are the right one to answer, because my sketch is based very much on your thread Serial Input Basics - Programming Questions - Arduino Forum
What datasheet are you reading?
The datasheet "Atmel-42735B-ATmega328/P_Datasheet_Complete-11/2016" from the link in the store http://www.atmel.com/Images/doc8161.pdf
I suspect there is a deal of luck in what you are doing. I believe the USART will receive a single character while interrupts are off but (I believe) that character will not be passed to the Serial Input Buffer until interrupts resume.
On side 233 starts the chapter "24.8. Data Reception – The USART Receiver".
There is written: "The USART Receiver is enabled by writing the Receive Enable (RXEN) bit in the UCSRnB Register to '1'."
Than in chapter 24.8.1. is written, "The Receiver starts data reception when it detects a valid start bit. Each bit that follows the start bit will be sampled ... and shifted into the Receive Shift Register until the first stop bit of a frame is received. ... When the first stop bit is received, i.e., a complete serial frame is present in the Receive Shift Register, the contents of the Shift Register will be moved into the receive buffer."
This sounds to me like at least one frame does go right into the buffer, without anything else but with the initialisation of the USART with "Serial.begin()" in "void setup()". But I'm not sure about it and that's the reason why I'm asking in this forum.
And based on that I believe that a second character would be lost. And if the data flow gets mixed up it might not be the 'R' that is received and processed.
But my handy app does only send the 'R'. That was my thought, that this one sign survives whatever happens because the USART receiver is always ready to receive just by hardware once it is activated.
A more reliable method might be for the Arduino to send an 'R' when it is ready for more data.
This could also be a way. What I wouldn't like, is that I should have to send periodically a "ready", instead of that the handy app sends a request, if it really has something to say.
I know nothing about the LED controller you are using or whether it really does need interrupts to be OFF for a long period of time.
The LED controller is from Adafruit NeoPixel and the library, the advice not to use interrupts when controlling it and the time calculations I'm using are from Nick Gammon GitHub - nickgammon/NeoPixels_SPI: Arduino library for NeoPixels
That would seem to be very poor programming.
"Eijeijei!" (German expression), somebody may get into trouble for this! 
I didn't want to mention my project again, because I did it before in this forum at another place. I was afraid, it would look like boasting, but maybe for more information about the background of my questions, here's again the link GitHub - orje/LEDviaApp_UNO: Control a LED stripe with the Arduino UNO over a handy app, programmed with the Quantum Leaps' QP™-nano active object framework, its QM™ modeling tool and the MIT App Inventor 2
I would like to get a definitive answer, whether one sign will arrive the receive buffer and stay there until I sample it with Serial.read() or Serial.available(), regardless of the state of the interrupts.
many thanks