How to tell if BCM replacement?

The car seems to idle low and want to stall, there is a piece missing that attaches from the exhaust manifold to the intake manifold and I would like to know what its called. I went to Auto zone and they had no idea.

Heat/Cold air doesn not blow out. Service wrench light, Service Engine Soon, ABS, and Air Bags flash on and off. Also, speedometer, fuel and temp guage erratic. I explained the signs to 3 mechanics. Clicking noise, ignition off. Diagnostics 2x. One said I needed an alternator ($578 for part/labor). The other a tune-up ($450 for Ignition coil/spark plugs and labor) due to misfires detected. Wasted hard earned money on their BS. Eventually, had a tune-up elsewhere at a much lower price. Problems above persist. The last mechanic, stated it could be car's BCM. If so, they don't deal with issue. Signs of BCM problem? What and Where should I go? Cost? I live in Phila, PA.

Here is a picture, passenger side by the firewall toward the front

Other pictures

No auto climate control. My 2nd Diagnostics/trouble shooting pulled up P0300 Engine Misfire Detected and PO305 Cylinder 5 Misfire Detected - not much else.

Thank you, I called the dealerhip and they said gm was out of stock and the dealershipe in the area, I went to parts. Com and I wasnt sure if the part I looked up to order was the right one

Prior to tune - up, mechanic pulled codes from computer dx. Test. AC/Heat, blower resistor shot. The rest, not checked. Again, could it be BCM causing erratic changes (Service Lights/Erratic gauges)?

Thanks for all your help. The part was there, I didn't see it. The tube moved up to screw it in, I didn't realize it. However, I took the battery out to charge it over night and now the car won't start so I don't know if that has something to do with it. I'm frustrated now so I'm getting it towed to the dealership. I'm thinking the new bcm does need to be programmed.

I can get in out car without a problem. Signals work. The warning signals/guages erratic. My fan doesn't blow hard (resistor shot). My AC button doesn't light up anymore. Car sounds like crap (loud) when I pull off, then settles down. Where should I go to BCM problem? If I don't get the problem fixed right away, what will happen?

It started as soon as I put the new bcm in. All the lights are on. It just chokes. The old bcm, it starts and then shuts right off

Should I take it to a dealership for service? It seems like the mechanics in the area don't have a clue when I explain the issues. What is the procedure to dx problem? Cost for dx and replacement?