Code p0671 trans slips occasionally and o/d light flsshes. Sometimes reverse doesn't work properly

First of all, Hyundai Transmissions use a 'specific' fluid called 'SPIII' and ONLY SPIII. Any other fluid doesn't meet Hyundai's specification and can cause 4th gear slippage (which is the TTC - Torque Converter Clutch - frequently referred to as the "lock-up" converter). You should have had a Check Engine Light. The problem you describe regarding 'doesn't drop back to first when I'm at a stop' sounds like the transmission has gone into 'Fail-Safe' mode because of the 4th gear ration incorrect. In "Fail-Safe" mode, the trans stays in 3rd gear until you cycle the ignition key off, and then back on. You need to scan your engine's computer to see what/which trouble codes have been set/stored in it. You'll need an OBDII scanner to read any codes that might be there. When a 'check engine light' (MIL=Malfunction Indicator Lamp) is turned on, there is a DTC (Diagnostic Trouble Code) set/stored in the ECM/PCM (Engine Control Module/Powertrain Control Module). This code identifies which engine management system component/function experienced a malfunction. Once you pull/read this/these code(s), you can begin diagnosis.

if the gears were "melted together" then the trans wouldn't shift at all. since the car runs fine sometimes and sometimes it doesn't it sounds like an electronic problem. the trans uses shift solenoids to acheive gear changes. is the fluid dark brown in color or smell burnt? if so you may have some slipping of the clutch discs/ bands, but it doesn't sound severe if the car runs fine at times. i suggest you have a trans shop to drive the car and get a more accurate diagnosis.

The VSC and trac will shut off with a P0500 or P0705 code, so diagnose those first. You left out year and engine, so we can't look up most common failures.

is transfluid at proper level if it is most likely check engine light is on and you should go to trans/ shop or dealer access fault codes

The trans probablely wasn't the problem.
Sounds more like solenoids or shift linkage issues

Do you have any codes for the check eng. light. I have seen the steering angle sensor on your model go bad which could cause the light to go on. The trans slipping could be your vehicle going into limp home mode because of the codes. Without running a quick test it's only a guess.
The problem with your gas indicator is the fuel pump, you will need to replace it. The card on the pump gets ruined from the sulphur in the fuel.
The problem with your lights could be the dimmer switch turned all the way down. Yes believe it or not this has happened.

U are going to need to overhaul the transmission here, the smoke is the trans overheating from excessive slippage i suspect.code for wrong gear ratio's mean the input and output shaft speed are out of symc for the gear the trans is in, the 3rd or 4th gear is the usual codes, this sometimes is caused by defective speed sensors but most times the trans is slipping and the TCM detects that and sets a code

There is a recall for the transmission that involves your exact problem. Call a dealer and see if your VIN is eligible for the recall. Check the NHTSA website for the exact details.

the same thing happened to my 97 sebring convertable.....i replaced the sensors and i still had the problem. take it to auto zones and have them read the codes from your computer. the computer will tell you whats wrong with it....the shift solenoid on my tranny went out and im still looking to repair it. the solenoid pack is 105-200 from ebay. but good luck finding the info to do it yourself. i been looking for 2 weeks and still nothing. good luck!

Do you also have a check engine light on as well?
Go to an auto parts store like autozone where they'll plug it up to a scan tool for the trouble code for free.
How do the forward gears react?...shifts rough?...clunks through gears?...slips? (engine speeding up but forward travel isnt)

Start on level ground...engine cooled.
If low or missing trans. fluid...any leaks?
Check for trans. fluid in the radiator (not the recovery). The radiator is a cooling unit for the fluid and in some cases can rupture , pushing fluid into the radiator itself. Check especially up towards the radiator area where the trans. fluid lines go into the radiator.

Is the access door for the oil filter in place? It needs to be.(has to do with cooling, and area of greater/lesser air resistense)
How about the fluid...smell?...dirty?...gooey?

For trans fluid ,check trans fluid level with engine on, hot and running in park, on level ground.

You have a 4L60E M30 series transmission. Has certain components (valves-solenoids) in it that can keep transmissions from shifting properly should the fluid overheat or the pressure become too high/low.
The usual trouble code this displays is P1870.
Sometimes it is simply a TCC valve out of whack and can be replaced