For all intents and purposes..NO, a loose rear differential should not cause a vehicle to fail an emissions or "smog" test.
The two systems are entirely: different, separate and have no common elements, or components.
The Emissions system, which controls the quantity of un-burned hydrocarbons (gasoline) in modern vehicles employes a host of mechanical, thermo-electric, and voltage sensing, components to send information signals back to the: ECM, ECU,..or most plainly put...the COMPUTER. The computer then interprets the Data, and adjusts various components based on info received from the sensors thru the computer. This keeps the vehicle operating at Federal Emission Standards.
A loose differential will cause: noise, vibration, wheel shudder, and possible steering problems, and if bad enough...eve
Loose lug nuts, failed CV joint, failed wheel bearing, failed transmission differential bearing.
Bad seal or loose drain plug.
The bus is the communication system between the computers on the vehicle. No bus means that the system has failed. Possible causes include broken wires, bad modules, loose connectors.The bus is the communication system between the computers on the vehicle. No bus means that the system has failed. Possible causes include broken wires, bad modules, loose connectors.
Improperly assembled, failed wheel bearing, loose lugnuts, wrong rim, etc.
There is a vapor leak in the fuel system. The most likely cause is a loose/failed gas cap.
back struts are bad or exhaust is loose
Sounds like the alternator has failed and you ran the vehicle until the battery was drained.
Weak battery, corroded or loose battery cables, failed starter.
A weak battery, loose/broken wires making a short, defective sensors that monitor what the gauges are displaying
No. A loose battery connector will cause problems drawing power from the battery to start the vehicle, and it may present a bit of a problem charging it (but that is not likely because the charging current is low compared to the starting current). If you have a loose connector and the battery goes dead, clean and tighten your battery terminals and start making some checks. The loose connector will not cause your battery to go dead.
A speedometer will stop working if the gauge has failed, the wiring is loose, or the sending unit has become defective. Each component must be tested to determine the cause.