You’ve done a pretty good job so far of diagnosing the problem. However, the results of the 0700/0741 combination will probably surprise you. It shows there is an electrical/electronic problem and not a mechanical one (by the way, diagnosing and replacing the bad range sensor will save you from other problems once you have the problem we are discussing solved.
With that said, a little background will keep things clear for you. In all transmissions built since the early 1990s (and before) there are little workhorses called control modules, whose function is to keep things running correctly and recording errors, working in the background. Unless one of the modules goes out of whack, you won’t even know they are there.
The control modules have other names that are probably more recognizable to you. For example, the Transmission Control Module (TCM) and Engine Control Module (ECM) are better known as microcomputers. They are the computers that control the transmission, of course, and all of the many subsystems that are required to meet the tight tolerances for emissions and mileage that cars and SUVs have to meet today.
In the transmission, there is one module, the Torque Converter Clutch Control Module (TCCM) that controls the torque converter. The torque converter is a device that balances the torque forces for the engine and transmission. In a manual transmission, you control the balance by accelerating and increasing the torque from the engine so that it matches the torque requirements of the transmission with the clutch and gearshift. The TCCM handles the same process in an automatic. It balances the engine’s output requirements with the transmission’s input requirements so that the vehicle doesn’t stall. Eventually, the Clutch Module takes over and locks up the transmission so there is a solid lock between the transmission and engine (similar to the top gear in a manual when you are finished running through the gears).
In an automatic transmission, there are a number of clutches (and bands) that engage as it goes through its gear train. In order to engage properly, a solenoid (Transmission Converter Clutch Control Solenoids) fires when the engine reaches the proper rotational speed (also known as revs or torque) so that the proper gear engages.
In cars of the 1990s, which were generally four-speeds, there were two solenoid actuator packs, one on the inside of the valve body and other on the outside. One of them controlled two gears and the controlled the other two gears (for example, solenoid-1, switch-1; solenoid-2, switch 1, first gear; solenoid-2, switch-2, solenoid-2, switch 1, second gear, and so on). Today’s transmissions (your 2007 was a five-speed, so it still relied on two packs as they could carry five gears) rely on multi-clutch packs and solenoids so that a six-speed, uses three solenoids, while an eight-speed uses four, and so on.
With all that as background, your Sportage has a problem with its solenoid packs. It could be with one or with two. From the sound of it, I think you have two bad solenoids that have to be replaced. Once you have repaired them, you will find your Sportage shifts and operates normally.
If you feel confident taking your Sportage’s transmission apart, then you can purchase both solenoid packs for about $300 and do the work, very carefully. Since you do have to tear down the transmission to get the valve body cover off so you can access, if you have the least bit of doubt, let a transmission shop handle the repair because they have the expertise to do it. It will cost you about $700 to $800 to have the repair done.
By the way, the reason your Sportage is sitting in second or third is because it is in limp-home mode. This enables you to get your SUV to a service area or home. The single-gear setting give you just enough performance to get where you need to go.
Please let me know what happens, I do want to hear it.