Where does the transmission fluid go?

Check the color of the transmission fluid, dark colored is bad. Overheating could cause the transmission fluid to go bad. Most transmission fluids are red and semi transparent. If the fluid is bad change the fluid, this can fix hard shifts and shudder.

not sure but you may have a leak in your oil cooler ,,,, this would allow trans oil into the coolant and cause your loss of oil and overheating,,,,

Could it be possible the coolant lines are reversed not allowing the trans fluid to reach the cooler? The pressure line from the trans pump can easily be identified by taking it loose and directing the fluid into a container.

check for engine block clogg

A few questions first...are you sure its overheating and not just a bad temp sensor? is yes then how many miles on car? if over 90K than did you have the water pump replaced with your timing belt?Is your fan coming on at all? Is your radiator full?If your water pump has been changed and your radiator is full and your fan is working properly and you have a new thermostat then you should take your radiator off and have it flow tested to make sure its operating properly, and no you cant just pour water through it to tell if its working properly.

Do the cooling fans work ?

Your transmission fluid is dangerously low (tranny is dry). The pops & jerks are the transmission
internal components trying not to seize onto each other due to lack of lubrication and/or overheating.
Low transmission fluid leads not only to loss of lubrication but lots of overheating (of the tranny).
The loss of Elantra performance (speed tops out at 55), is caused by power loss due to friction inside the transmission. All your power is lost overheating the transmission (toward destruction), instead of
being transmitted through the transmission to the wheels.

The 2004 Elantra takes Hyundai SP-III Automatic Transmission Fluid, with 8.4 quarts used in a
total refill - but just 5 quarts at a fluid change. For How-to transmission fluid change instructions:

http://en.allexperts.com/q/Hyundai-Repair-815/2004-Elantra-GT-transmission.htm

www.autozone.com claims that 2004 Hyundai Elantra can take these non-Hyundai ATF's:
Lucas Semi-Synthetic Multivehicle ATF
Pennzoil Muilt Vehicle ATF
Valvoline Maxlife Dex/Merc ATF
Castrol High Mileage ATF

have you try changing thermostat that causes it to overheat sometimes if you have a leaky hose get it changed i have a 96 corsica i had the same problem fixed hose changed thermostat still was overheating took to a radiator shop they tested it it ended up being that get it tested first if they tell you it is the radiator they are probably gonna charge you A LOT take your time and doit your self if you think you cant do it then once you know is that look around to see who charges you less.

you probably have a stuck thermostat

Here is a shot in the dark, without knowing year , size engine, or transmission. But it sounds like you may have transmission fluid in the cooling system, If so the transmission cooler in the radiator may be bad. Just a thought.