The conditioning chip as you put it is that big black blob. It is basically a microcontroller. In order to save money and space, the silicon itself is soldered directly to the board (rather than a seperate IC). It is then covered with a blob of epoxy to protect it.
I can tell T2 is GND because it is part of the ground plane (the large filled region). If you look at the second photo, you can see there is no gap around the edge of it, meaning that it must be connected to GND.
I can also tell that T5 is Not GND because if you look to the 'south-west' of the test point (assuming 'north' is up), you can see a trace coming from it and joining the trace to pin 6 of the EEPROM.
You have two options for joining it up. The first is to look under the screen and underneath you should find a series of solder joints - these could be desoldered and then have wires attached which go to the Arduino (i would suggest a 100Ohm resistor in series with each of the 16 wires until more is known about the display - just a precaution).
If it is not soldered on, but rather attached by a rubbery looking pink and black strip, then things get a little more interesting. You would in that case likely have to cut the traces going to the blob and scrape of the solder mask (using the back of a knife) and solder your wires to the cleaned PCB traces.
As for probing stuff, an oscilloscope is handy if you have access to one, but is not a requirement by any means. You can do some probing work with just an ordinary digital multimeter (even a cheap £5 one is a handy tool to have).
For digital signals, you can use an Arduino as a digital probe - I found a great site which has an arduino sketch and computer software which works well - I will have to find the link.