7400registerttl
First of all, I don't know if this kind of question is allowed here, so sorry if it isn't.
I am looking for a 7400 series register which has common in and out pins (like for the RAM ou EEPROM: 
) with a load signal and a store signal. I searched on Google but I didn't found anything (maybe my searches are wrong).
Does it exist and if yes, what is its name/reference?
You could use a MOSFET driver such as the TC4427 with a 47Ω series resistor. The resistor will sufficiently limit the power to protect the output.
Note that with a 50Ω output resistor the voltage on a high-impedance input will be double that on a 50Ω input.
What you really need depends on the device that takes the signal as input.
The circuit you implemented will work properly for all the cases you stated except for the last one. For S1S0 = 11, it will perform shifting and serial data loading.
For parallel data loading, you must connect the I_3 lines of your MUX to the parallel-data-in lines (4-bit).
Image courtesy: @IANIK
Best Answer
There isn't a register that has common input and output pins, but you can certainly wire up a '373 or '374 (8-bit latch or register with tristate outputs) to achieve that functionality:
simulate this circuit – Schematic created using CircuitLab