The only way you could get the readings shown in your photos is if you removed the 1 megaohm resistor you had before as a voltage divider and are using the sensor by itself. If you do the math , taking into account that your NO LOAD is 0 volts and your MAX LOAD is almost 5V, that means that the resistance goes very low pulling the voltage up. You must be using the center pin for your readings and have the other two pins of the strain guage connected across the +5V.
If you draw a voltage divider and show the circuit you linked from the Sparkfun page, that means that with no load the sensor resistance would equal the 1 meg ohm resistor and the output would be half the supply voltage ( 2.5V). Your photo does not show that.
When I connect the multimeter to the pins the value on my LCD directly sinks from approx. 60 to 34 with no load and the Multimeter shows 0.00 VOLT. (Knows everyone why the Value on the LCD directly sinks??)
There must be something wrong with your wiring or the way you are connecting the meter and lcd. The meter SHOULD NOT effect your reading on the lcd. You need to assume that there is a short somewhere with the lcd. Remove the lcd and read the value on your laptop with the serial port so you can see the value measured by the arduino with NO LOAD.
Make sure the LCD is completely removed from the circuit before taking this measurement. You have just the sensor and the arduino, nothing else. Then measure the value with the meter. Do not use alligator clips. Use a solid copper 22 guage wire as a jumper for the arduino to meter connection (or a pre-made breadboard type jumper). Take a photo of the meter with NO LOAD and NO LCD. You 0V reading is bogus. There is no way that can be correct.
Draw a schematic of how you had everything connected BEFORE you added the connection to Mains ground and AFTER you added the connection. There was something wrong with the way you had it wired before and there is still something wrong with your connection to the LCD. What is controlling the lcd and where is it getting it's power ?
What is controlling the LCD ?
s it a possibilty to use an "amplifier" as an MCP6001 or something else?
Yes, you can use the circuit shown on the datasheet but before you do that you need get some good meter readings with No load .Your previous reading of 0 volts is impossible. You need to measure the NO LOAD voltage WITHOUT the LCD connected.