Multiplexer CD4051B gives strange readings
I'm using a breadboard of questionable quality.


That is what seems to be the source of your problems. Problematic breadboards cause loose connections within the board which you cannot see and hence lead to situations such as open connections etc. I'd try using a different board. 

Also the pulldown resistor need not be that high. A 10K resistor works well in most cases. I'd also try switching the IN value from the Arduino instead of connecting it to VCC. I haven't seen the block diagram for the chip but it seems that you may be shorting something internally inside the chip which is why the input drops to 0V after a short time. At the very least, connect a pullup resistor of about 10K between IN and the 5V power source. 

Though this may not be very relevant in this case, its important to note that the 4051B is a CMOS device and hence anything less than 1.5V on the input is 0V on the output and the lowermost threshold for a HIGH input is 3.5V. This means that this chip will not work with 3.3V devices without appropriate voltage level translations.