that little rectangular thing is the battery "intake" tube, so to speak. It vents the H2 gas from around the battery. It should be inside of the shroud of the battery, not sticking out.
Also, your positive battery terminal is disconnected.
To probe you o2 sensor, you need a HIGH IMPEDENCE MULTIMETER.
somebody please correct me if I´m wrong, but I think this is right:
Probe the o2 sensor by putting the POSITIVE lead of the multimeter up to the o2 wire (you can find a connector, follow the o2 sensor wire up to the distributor area) and then put your NEGATIVE multimeter wire on the CASING of the o2 sensor. DO NOT PUT THE NEGATIVE MULTIMETER WIRE ON THE NEGATIVE TERMINAL OF THE BATTERY, IT WILL "SHORT OUT" YOUR O2 SENSOR AND DESTROY IT.
Then you run the car, let it heat up (the o2 sensor must be heated by the exhaust gasses since it´s not a heated sensor) and then take readings. It ranges from 0-1V , 0 being lean and 1 being rich if I´mnot mistaken.
If your car runs good with the new filter don´t bother probing your o2 sensor. Just a waste of time. If putting a filter on the car messes up the a/f ratio that bad there is something seriously wrong anyway. The VAF sensor will take into account more air going in which will adjust the a/f ratio so don´t worry about it.