Is it Possible to Control Vehicle Key Fob/Remote Start with arduino board?
All of the Remote Starter systems I have seen are aftermarket products. In fact, most auto dealers use these after market products when installing them in new vehicles. It should be easy enough to find the manufacture of yours, and send for a replacement remote fob. Once the fob is programmed to your vehicle, you can hack the contacts to your new circuit. 

As far as sensing the engine running
most of these units use a sample of the engine RPM (tach) and detect the pulses. The RPM outputs on the ECU of the vehicle, and is probably already tapped to use for the Remote Start unit. For the report back function, you can buy a remote  fob &amp; receiver and hack it in much the same way as your starter key fob.

EDIT:
After reviewing the new information and pictures, I will revise my answer to the following:

If the led in fig 1 (lower right corner) is the confirmation of a start indicator, then a photo transistor can be optically coupled to this led through something like a short hose or tube. This can then be an input of the Arduino. 

EDIT 2:
In fig 1, middle of the board may be a tactile button switch to initiate a start. If this is the case, a relay could be connected to the Arduino output and wired to the tactile switch in parallel.