I just removed my cat convertor (mine is a Japanese RX7 so no pre cat; just cat) and replaced it with a straight ss pipe. Now I have these doubts.

1) Besides the sensor on the right, there was a pipe going into the cat on the left. What is that ; the air pump pipe?. I made a hole in the straight pipe and put it in. Is that correct?

2) I am hearing back fires now. Is that a bad sign? The worst one I heard was when starting the engine.

 

1) The pipe going into the cat is from the airpump. You don't need it connected to your mid pipe, you can simply tie it up out of the way. But since you already made a hole, I guess just keep it on. Since you no longer have a cat you can disconnect the grey power connector on top of the airpump (effectively disabling it). Now you won't hear that awful "mooing" you may have heard before.

2) Are they little back fires or big, loud ones? I hear small ones occasionally. I consider it to be the norm for a straight-through exhaust, as you've eliminated back pressure. But that's just my theory.

 

It is very normal to get back fire when you have removed the cats because you have removed the afterburning the cats are designed to do.

The cats will burn most unburnt gas by a heat exchange core that becomes heat soaked to the point of ignition. The Air pump adds more fuel to the fire by providing O2 to the mix. I could see the addition of O2 by your Airpump into your straight pipe could cause a cannon effect becuase you hvave given the fuel something to burn with. You may get less if you disable the air pump as described above.