Welcome to the forum!
The DPFE can fail, but generally symptoms are very minor and it won't cause a no-start. Does sound like yours has failed at some point. They are very easy to replace and don't cost too much, about $40 on RockAuto.
You can try swapping just the DPFE or checking for electrical faults related to it such as wire damage, but I doubt that will fix your no-start. The code 186 sounds like a clue that your problem is fuel-related.
Check your MAF; I wouldn't suggest throwing parts at it, but see if it runs any better (or starts) with the MAF unplugged. Unplugging the MAF forces the PCM to use speed density and guess at the necessary fuel to inject for what air it thinks is coming in. Sometimes this will work if your MAF is truly bad.
Check fuel pressure, this I am thinking is more likely. A fuel pressure test gauge is available at most parts stores. It screws onto the test port on your fuel rail (no parts removal required) and reads the fuel pressure at the rail. When you turn the key ON (without starting) the pump should run momentarily and prime (pressurize) the system to about 30-40 psi. This pressure should stay and take at least 20 minutes to leak down. This tests that the fuel pump is putting out enough pressure to prime the system, and that the fuel pressure regulator is regulating the pressure to the proper level and holding it tightly instead of leaking. It also tests if the injectors are closed, as they should be, and holding pressure. Once you start the engine you can read fuel pressure the same way; the reading will fluctuate somewhat but you should see the pump keeping up with demand.
The code 186, Injector Pulse Width Longer Than Expected, indicates that the PCM is commanding the injectors to stay open for an abnormally long time. The PCM will try to feed the motor enough fuel, but if fuel supply is too weak, it has to open the injectors for longer and longer to try to cope. Or, if the PCM is erroneously told the engine is running lean (too much airflow is reported due to a malfunctioning MAF) it would dump extra fuel to try to compensate for what amount of air it is told is coming in, and it will run the engine rich. I doubt the latter is your issue; it sounds to me like a failing fuel pump. The PCM has no way of knowing that the pump is failing and simply tries to open the injectors longer to feed the engine enough fuel.
If the pump is truly to blame, never fear, it is not too terrible a job and does not require dropping the tank unless you want to. You can cut an access hole in the body and change the fuel pump through there.
Before throwing any parts at your problem, such as a MAF or fuel pump, please try the fuel pressure test and MAF disconnect that are described above, and let us know how it goes.