I finally found time to do your very helpful scientific experiment.
Was to complicated for me. 
But for anyone else having this problem:
My z axis minimum was set to z0
Changing M208 for Z0 to Z-2 in config.g was finally solving the problem.
@atall thank you all for your great and quick help, tips and tricks.
After 10 months of guessing and time consumable troubleshooting I am finally able to use my most expensive printer!
hooray!
my setup:
I have assembled the printer myself. The parts are printed by myself.
Still no change.
Thank you for your help in advance.
Thank you very much for your answer.
In deed my extruder was a little bit loose (compared to my einsy prusa extruders). I have completely diassembled my extruder on my caribou duet and retightened all the screws an cables.
Now my extuder is mounted rocksolid. But I still have a poor first layer. 
My first Layer is my USP. And my duet printer is my most expensive one. So achieving a perfect first layer would be great.
yes, there is always a heightmap.csv generated.
I have included in the start gcode of my prusa Slicer following lines:
"
G29 S2 ;clear all stored mesh
G29 S1 P"fakeheightmap.csv ; use the "Faked" heightmap
"
The result has been as expected. In the middle I still have a + 1mm offset in my fakeheightmap.csv. My first Layer looks exactly as usual. But you can see that in the middle the contour line has way less contact to the printsheet as to the right and left.
As a result my heightmap.csv is not ignored. My heightmap.csv ist simply too "weak"?
I will try what @martin7404 suggested. I will use a gauge and measure myself to compare the measured values with the BLTouch ones. I hope that helps me to get a clue of what is going wrong here.
Are there any known issues?
Thank you for your help!
Hello @oliof ,
thank you for your beautiful idea!
I modified your idea slightly. Instead of changing only one (of 81) point(s). I changed a complete line.
But hooray. This method worked for me. I was able to see the rise and fall of the nozzle in the middle of the bed, where the fakeheightmap.csv was active!
At least I know now that there is some heightmap usage in the Background.
but sadly I still don't know what is wrong with my heightmap.
Could it be a bad offset of my BLTouch?
Or any other guesses?
Thank you very much in Advance! 
Hello @martin7404,
thank you for your answer.
G29 S0 did not wirk for me. I have just tried to implement it in the Start code in prusa Slicer.
The mesh measuring on all 9x9 points was made according to config.g.
But: the contourline of my print looks as bad as before.
On the left the nozzle is way to close to the bed. On the right there is nearly no contact between the nozzle an the bed.
My bed is too close on the left but too far on the right. So I assume there is absolutely no mesh compensation going on.
Thank you for your help!
The start.g attached to my first post is from a friend of mine. He owns the same printer. It is working for him. When I apply the following start.g (the "official" from caribou3d) the first layer is the same:
Picture of my heightmap.csv in the systemsfolder (9x9):
gcode-file I try to print (same result when I use prusa slicer 2.3):
; generated by PrusaSlicer 2.4.0+win64 on 2021-12-30 at 16:47:56 UTC
;
; external perimeters extrusion width = 0.60mm
; perimeters extrusion width = 0.50mm
; infill extrusion width = 0.50mm
; solid infill extrusion width = 0.50mm
; top infill extrusion width = 0.45mm
; first layer extrusion width = 0.42mm
M107
;TYPE:Custom
; =========================================================================================================
;
; start script for PrusaSlicer for CaribouDuet
;
; =========================================================================================================
;
G28 ; home all axis without mesh bed level
G0 X60 Y-3 Z80 ; move extruder above bed, keep extruder in front for cleaning and checking
;
M104 S160 T0 ; pre-heat extruder to 160°C
M140 S60 ; this will take the layer 1 temperature for bed 0
M190 S60 ; wait for bed temp
;
G29 ; mesh bed leveling using defined mesh grid
G0 X0 Y-3 Z0.6 ; go outside print area
;
M104 S215 ; set extruder temperature
M109 S215 ; wait for extruder temp
;
; =========================================================================================================
;
;
M98 P"0:/sys/primeLine.g" ; execute primeline macro
;
G92 E0.0 ; set extruder position
;
M572 D0 S0.07 ; set pressure advance
;
; =========================================================================================================
G21 ; set units to millimeters
G90 ; use absolute coordinates
M83 ; use relative distances for extrusion
M900 K0.05 ; Filament gcode LA 1.5
M900 K30 ; Filament gcode LA 1.0
M107
;LAYER_CHANGE
;Z:0.2
;HEIGHT:0.2
;BEFORE_LAYER_CHANGE
G92 E0.0
;0.2
G1 E-.4 F2100
G1 Z.4 F10800
;AFTER_LAYER_CHANGE
;0.2
G1 X4.516 Y7.168
G1 Z.2
G1 E.4 F2100
M204 P1000
;TYPE:Skirt/Brim
;WIDTH:0.42
G1 F1200
G1 X5.452 Y6.976 E.02998
G1 X245.296 Y6.928 E7.52013
G1 X246.603 Y7.32 E.04278
G1 X247.48 Y8.365 E.04278
G1 X247.673 Y9.305 E.03008
G1 X247.673 Y49.309 E1.25429
G1 X247.426 Y201.94 E4.78564
G1 X247.032 Y203.246 E.04278
G1 X245.985 Y204.121 E.04278
G1 X245.051 Y204.313 E.0299
G1 X5.274 Y204.483 E7.51806
G1 X3.967 Y204.092 E.04278
G1 X3.09 Y203.046 E.04278
G1 X2.896 Y202.106 E.03008
G1 X3.076 Y9.35 E6.04373
G1 X3.469 Y8.044 E.04278
G1 X4.47 Y7.207 E.0409
; printing object Form-Kubus id:0 copy 3
for my taste (taste because I am a noob and have no knowledge 
) the g29 Sxx is applied a little to often. Is there any known interference?
I have often read that possibly the z-Motors don't get enough voltage for mesh compensation?
I have already checked that my extruder can travel along the gentry easily. A tilt of the gantry can be performed easily by turning one spindle.
I have not found any further (gcode-)commands where the used mesh can be visualised? Is there a way? Or is there a way to easily visualize the motor signals? So that one can see if there is a signal for the motor to turn?
...and again thank you for your much appreciated help in advance!
Good evening,
I am completely helpless. I am trying to get my caribou duet working. This prusa style printer is equipped with a BLTouch-Sensor.
As stated in the headline whilst printing the heightmap is ignored.
Entering M122 while printing the printer says mesh is enabled.
My first layer is definitely printed without any mesh compensation (visible).
What I already tried:
I have absolutely no further clue what could be the possible reason for my printer not to use the heightmap and compensate my printing bed.
Thank you for your help in advance!
Required files are attached.
DWC3.3
Board: Duet 2 WiFi (2WiFi)
Firmware: RepRapFirmware for Duet 2 WiFi/Ethernet 3.3 (2021-06-15)
Duet WiFi Server Version: 1.26
start.txt PrusaSlicer-StartScript.txt homez.g heightmap.txt config.g bed.g