Wonder why my car 's engine vibrating/ shaking a lot at idle speed. Is this normal for this year model with 4 cylinders or it's really an engine problem ?

More information please. I am assuming this is a Toyota. What year is this vehicle? What model is this vehicle? I am also assuming this is the 1.8 litre, four cylinder engine. The most common firing order on a 4 cylinder engine is 1, 3, 4, 2.
Does this engine idle smoothly when in Neutral and in Park? When in Park/Neutral gently press and hold the gas pedal at a slightly higher engine speed. You don't need to scream the engine just push it up to its normal working speed. Does it still run smoothly? If the engine idles and runs smoothly it suggests that it isn't the firing order causing the issue.
Do you have this problem when you select Drive or when you are actually moving? Your best option here is to take the vehicle to a good mechanic that way they will be able to see, hear and feel the vibration. This will help work out what the problem is.

Depends on the engine. Really need year make and model

Yup, you said it yourself, Your a parts replacer and not a mechanic. I understand friend. Your engine problem is the fault of a bad or really dirty fuel injector. Take it to an Auto Zone and ask them to check the codes on your vehicle. The code will read P0304 if it's # 4 cylinder that's misfiring. We call it a dead cylinder that doesn't seem to always be dead. Because normally it just shakes a little at an idle, is smooth going up the gears, until it gets into overdrive and it will shake and vibrate like all is going to come apart. Find out which cylinder is dead and change that injector. Done.

you might have a broken motor mount or ****** mount,as for the overheating you might need a new thermostat or water pump,but check your antifreeze first to make sure its strong enough for this time of year

The most common cause of this is that the engine mounts are hard from age and engine heat and no longer can absorb the normal vibration of any 4 cylinder engine. Also the idle may be slow due to a dirty idle speed control system or the engine can need a tune up or has dirty fuel injectors, I would service all these to correct the problems.

engine shake could be misfiring or even a broken engine mount. Not related to your battery.

Sounds like it is missing badly and would not be so noticeable when running. Have you had it tuned up lately? Could be wires and plugs need replacing. Also if you have a lot of miles on the engine the compression in the cylinders may be low. Two possibilities.

I've got an 08 4cyl and it doesn't really have the vibration unless its really cold. Usually if I give it about 30 seconds...when the rpms start to come down, its barely noticeable.

The vehicle is going to idle higher when the a/c is on to compesate for the extra load that is on it at an idle.Dont worry about it.

dude its normal for first start or cold start that the idle is ok but it must high than normal... its normal because the engine is warming up it self... when it gets warm, the idle speed will return normal idle... your engine is 1200cc, it has not much HP for a great load... and i know its a carburatored engine and its an old engine, your head lyt and a/c fan and aux fan and all the electrical system on your car, uses the HP of you engine and its a lot of load to the alternator driven by the engine to generate an electricity they need and thats why the engine rpm speed lowering down to dead and its shake. thats why when you use your lyts or all electrical, you engine is weakening... the best remede is adjusting you idle speed... its located behind the carburator.. its a small black knob.. just make a twist each eighter way and you'll find out if your idle rpm speed ts incresing or decresing.. just increase it and try opening your lyts and other electrical, if its still shaking then adjust it a little more