Engine shakes

It doesn't do either of those. It only shakes (not often but rarely) when I turn the key to turn the car on. Once the car is actually on it never shakes. I never actually said that the engine shakes. I don't know that. I've just felt a shaking from time to time. It could be the engine or it could be something else. If this is normal I would be pleased because, as I said, I've just spent $3,200 on my car in the past month, give or take. It's not a big issue. Just something that I wondered about.
Thanks

Engine Mechanical problem
2004 Hyundai Elantra 4 cyl Front Wheel Drive Automatic

the engine is 1.6L, recently skim it's head cylinder due to overheating.
Before resurface the engine head, the engine condition is normal however there is oil leakage around the head and block contact area. To resolve the problem I resurface the cylinder head block to get the flat surface again. However after putting all back together, the leakage stop but the engine shake abnormally during idle.
The spark plug in cylinder 2, 3 and 4 become darkish like it did not have a normal burn environment.
Have sent the car to check it ecu for error code problem but there is none. (The check engine did not light up initially until now)
what should I do now? What more test did I need to do?

-there is no error code display in the diagnose test
-timing belt are install correctly according to its marking
-valve clearance are not adjustable

-what is the normal compression value for the cylinder?
I will post the value as soon as I can.

At first test, pressure at cylinder 1 did not have any pressure, while the other 3 cylinder have normal pressure.
So I decided to remove, clean and re-install the cylinder head. But now the problem seams to shift.
Only in cylinder 3 have pressure while the rest i.E. Cylinder 1, 2 and 4, having lost pressure. I try pour some oil inside the 3 problem cylinder and then re-test the pressure. Only cylinder 1 I could obtain back the pressure while the other two still having the problem.
In short, before:
cylinder 2, 3, 4 = ok
cylinder 1 = lost pressure

after:
cylinder 3 = ok
cylinder 1 = ok (only after pour some oil)
cylinder 2, 4 = lost pressure

do have any suggestion on what to do next? Why the problem shift from cylinder to cylinder?

2005 Hyundai Elantra Engine short rapid vibration shake felt in seat and steering wheel equally while car in gear at a stop but not in Neutral or Park position, vibration/shake worse at higher speeds, ruled out tires, rims, ball joints, hub bearings brakes.

Everything done to vehicle so far:
Checked ball joints, tie rods, hub bearings for any play and found nothing wrong. Front and Rear alignment performed, Replace 2 warped rims with more than 1/8” run-out that were causing severe vibration at speeds above 50mph, Installed new tires, road force balance all 4 wheels/tires, re-surfaced both front brake rotors due to both slightly warped. This has taken care of 50% of vibration problem rapid shaking/vibration still there at all speeds, worst is felt at 55-70mph.

I feel the shaking/vibration in the seat and steering wheel, most noticeable in the seat. You feel it the best on smooth pavement. When I put the car into any gear while brake applied at a stop the engine shake/pulsates gets work as you start to accelerate forward this is the same feeling shake/vibration felt at higher speeds. When transmission is placed into neutral or park the shake significantly decreases and can hardly be felt. I believe the cause of the problem to be an engine/ transmission related problem like a motor mount or mount on left side where transmission is mounted or something on chassis/ frame is loose.
Please help, I have already paid $919 for replacement parts and repairs.

I did not understand the 'lifter' that you mention.
Is it the short, cylinder type (look like button) located between valve and the rocker arms?
Would you mind provide me with some visual aids or the location of this 'lifter'? It is sure will help me a lots.

I have done what advice that you guys give and it seams that the car is going to be fine. There is still some knocking at the engine but as you say before, the noise will fade away. I hope it is true. Until then, if there is any more problem I will consult to you more. As for now THANK YOU for all the advice, figure and diagram that you gave me. It is sure make it easy to solve it.

Hello, I have a elantra 2005 with 15000 km mileage. The engine vibrate badly while stationary state. When I push the acceleration a little ( under 1000RPM) engine shakes badly. The mechanic has checked all the engine mounts but they were well. I have also changed the spark plug. The mechanic has checked the engine with computer but he didnot find anything. I hate this vibration, what should I do?

I will do what I can. Thank you for your help so far.

The thing that confuses me is that it only happens when I turn the key in order to turn the car on and it doesn't happen most of the time. Is this ever normal?

When I look back I seem to remember having a similar issue last year and when I took it to the dealer I found nothing. I never noticed this issue for 3/4 of a year. Now I have noticed it again (but only on a few starts.) I guess the only thing to do is to take a video if I am able to do that. I am not good at posting things to youtube. I'd have to figure it out. I hope it's nothing. We will see.