Based on that picture the pink foam is just there to rest the display on to stop it bouncing around and scraping against the board. If you remove that, you will probably find solder joints underneath it - that would make life easier.
Soldermask is what makes the board green. It is essentially a layer of high temperature paint under which are the copper traces. If you scrape off the paint, you reveal bare copper that you can solder too - but that is a last resort if you can't find solder joints under that pink block of foam.
As for setting to 0, you could probably wire an arduino pin directly to the tare button and then use that to trigger the scale to reset.
It is possible that the display could be DC. It looks a similar design (clear transistors sandwhiched between two layers of glass) as the one in my alarm clock. That one is connected directly to the outputs of a what appears to be a PIC microcontroller.
If you can find solder joints, I would suggest turning the scale on, setting it to 0, and then using the multimeter between each pin of the display and ground to see what it looks like. (You can use T2 testpoint as the GND terminal of the multimeter).