Measure/Access Information from small device through audio jack (mic)?
Grabbing the display signal doesn't seem to be feasible because it are multiple digitally coded lines and not one signal with amplitude. 


LCD waveforms can be translated into data, but it requires some time and effort.  Generally the display is static, so you can use an ADC and a system of multiplexors to check each segment and common in turn and it can be relatively easy and fast once you understand how simple segmented LCDs work.  As a beginner, though, it can seem daunting.


  Could I maybe listen to the measuring signal through the audio line and then try to do my own reference measurements, or might this disport the actual measurement?


It completely depends on the medium and the measurement process, but most likely you will alter the reading by tapping into those lines.


  Is there maybe some other way to grab this information to provide means of interoperability?


You should probe around the board with an oscilloscope during and after a measurement to see if the designers left a serial stream or other data stream coming from the device for debugging and development purposes.  If you can identify the chip or circuit, you may be able to find enough information about it to make some intelligent guesses as to how it might send data out other than the LCD, if it does at all.


  What voltage (&amp; current?) can be measured my a mic in line and how must the signal be modulated? Some fixed frequency or can microphone electronics also analyze frequency?


I wouldn't put more than 250mV voltage swings into the microphone of a device I cared about.  Additional testing will help you find an appropriate input level, but I'd start off small at around 10mV, fed through a 470nF capacitor into the microphone input, then go from there to find the dynamic range of the microphone input of a given device.

You might find more help if you told us what the device is and does, and showed pictures of the internal circuitry and measuring apparatus.