Welcome to the forum! Try using the "Search this forum" feature from the forum main page for keyword "P0171". You'll find that these codes have been addressed several times on this forum along with some good descriptions on how to check the PCV elbow, intake manifold, and comments regarding the DPFE. Since you have the code for both banks of cylinders the issue is likely to be something related to both banks such as the PCV elbow. An intake manifold/gasket would likely start out impacting only one bank.
The P0306 is a misfire code for cylinder #6. If it's been awhile since the spark plugs and wires have been changed the source of the P0171 and P0174 may have caused a plug to foul. Or you might also have a fuel injector that is not functioning properly contributing to the raw fuel smell.
If you plan to keep the vehicle for awhile you might start with a tune up (spark plugs, wires, fuel filter, air filter) since that is routine maintenance and could solve all three issues, but I suspect you'll still find yourself searching for the other issues addressed for P017x codes.
Have you cleared the diagnostic codes and checked to see if the P0171 and P0174 came back? It would be difficult over the Internet to distinguish between the PCV and the manifold gasket just from the symptoms and codes. Determining which is required will require inspection. I would expect the shop has the capabilities to diagnose which is the more likely culprit, or if nothing else, which one it is not.
I have driven the E about 60 mile today. Here what I see so far:
The check Engine light is OFF. Is the code available when the like OFF?
The idle is shaking.
Power is still losing (but not feel much shaking). I can speed up to 60 mile but not quickly accelerate like before.
No more fuel smell
So time heard straight click, click .... noise (about 5 minutes) seem like from valve on from the back of the engine. and then it gone by itself
The ticking may be associated with a plug misfiring, that would explain the residual shake and lose of power.
I would remove all the "NEW" plugs, use a thread chaser(particular attention to #6, close to firewall, driver side), regap the plugs, use anti seize, reinstall plugs. Check the wires, replace if defective.
Pending codes could still be gathered even if the Check Engine Light (CEL) is not currently on. Sounds like you still have a vacuum leak or ignition misfire. You might want to make sure none of the spark plug wires have come loose from the coil or plug.
Remove spark plugs 1, 4, 5, and 6 (2 other need to take off the wheel). #1, #4 and #5 are white like new. #6 is black and fuel wet.
Install the plugs back, replace #6 wire still same issue (shaking and lack power)
Disconnect #6 from the coil, nothing change like #6 does not affect any thing
Lift the #6 wire close to the #6 coil, I have seen sparks between the coil and the wire. Is that mean the coil OK?
Inspect vacuum lines, I did not found any thing
What should I do next? replace coils, PCV valve, fuel or replace intake manifold gasket?
I suggest you remove the plug from cylinder #6 and clean it, then swap it with a plug from another cylinder. See if the misfire follows the plug or the stays on cylinder #6. If if follows the plug then you probably have a bad plug. If it remains on 6, then try swapping the spark plug wire with another wire of similar length.
If the misfire stays on cylinder #6 and the spark plug and spark plug wire have been swapped with different cylinders that would suggest you either have a coil issue, a compression issue, or a fuel injector issue. Since the plug was wet with fuel like you mention in post #9 above, the injector is probably not the issue unless it's stuck open all the time.
You should use a spark tester to confirm that there is spark to the plug for cylinder #6. You don't want to just let the plug wire arc to the engine or body, that is hard on the coil.
Finally, I think it is time to get a new car.
I have changed intake manifold gasket and PCV valve. The car have improve power and vacuum, but still shaking . I still heard some metallic noise from the engine as mentioned in previous post and then the noise gone. I let the car in idle for 30 minutes and then took drive test about 10 miles. The power have some improve but not as it was. When driving into garage, I heard strong noise like some think broken from the engine and the E totally dead. Try start again but no luck. Inspect engine oil from the oil cap, I have seen some metal powder in oil and the cap I think some think have broken in the engine or the head. This problem is over my hand.