The code is still stored even if the light goes out by itself. As stated, take it to an auto parts store and get the code. Then you will know what needs to be fixed.
From the owners manual:
18. Temperature Gauge
The temperature gauge shows engine coolant temperature. Any reading within the normal range indicates that the engine cooling system is operating satisfactorily. The gauge pointer will likely indicate a higher temperature when driving in hot weather, up mountain grades, or when towing a trailer. It should not be allowed to exceed the upper limits of the normal operating range
You can buy code readers for less than $20 these days. Or, drop by an auto parts store and they can read it for you. Yes, it's a sensor causing it and you need to know the code to know what sensor it is.
You're kidding right? Transmission is not a "throw away item". As stated, try an actual transmission shop. Not sure if the transmission used with the diesel is the same as the one used for the gas engines. Do you have an owners manual?
You can take it to a wheel shop to determine if it can be repaired, otherwise, as stated, if it holds air and you don't notice any new vibration, it's still good.
Could be a module in the transmission, or a bad connector/wiring or a serious internal issue. If you are a DIY type of person and have the means, drop the pan and do a fluid/filter change.
It's really hard to tell over the internet with certainty. Did you check the fluid level properly? What's the mileage? You can start by getting the transmission serviced. Doubt adding a "small amount" of fluid would accomplish anything. You really need to take it to a trans shop or dealer for a proper diagnosis.
The best tire height would be one that does not exceed factory width/height so you won't run into any fitment issues. There are several online tire size calculators you can use to enter your current size then play around with the 20" sizes to get the one that matches closest in width and diameter.