Unfortunately, it looks to me like you are ignoring the most significant symptom that you reported, namely, "I saw a little steam coming from the front". Doesn't that sound to you as if the engine was overheating?
If the engine was indeed overheating, then we have the question of how long the engine (AND the transmission) were subjected to temperatures that were too high. You did not mention the reading on your temperature gauge, so I am assuming that you did not look at that gauge.
And, then, we have the issue of your next step, namely, "I turned the car back on to run the AC for a few minutes". If the engine was overheating, restarting the engine after only a very brief time was a very big mistake.
How does the issue of engine overheating relate to the transmission fluid? Simply because you could have "cooked" the fluid by driving the car while the engine was overheating. In addition to possibly having cooked the transmission, significant damage to the engine could have resulted from your actions.
When you observe steam coming from underneath the hood of the car, and/or if you observe a very high reading on the temperature gauge, the only appropriate thing to do is to pull off the road as soon as you can safely do so, shut down the engine, open the hood, and allow everything to cool down properly--which could be as long as 45 minutes. While the hood is open, you can assess whether there might be a broken radiator hose or a leak.
In truth, restarting an engine that appeared to be overheating is very unwise and under those circumstances I would recommend having the car towed in order to have the problem resolved prior to retarting and driving the car. Overheating engines do not suddenly cure themselves, and the origin of the problem needs to be found prior to driving the car again.
As to the alternator, that failure could simply be coincidental. Yes, 3 years/79k is early for alternator failure, but early alternator failure can happen.
At this point, you need to have the trans fluid changed before operating the car again, but you also need to have a competent mechanic examine the cooling system for the cause of the apparent overheating. He should also perform a compression test in order to determine if you warped the cylinder head as a result of the overheating.
When was the last time that the coolant was changed? Do you know for sure that there was sufficient coolant in the radiator? What is the maintenance history of this car?
We really need to know a lot more about your car, but in the meantime, try to remember to never drive (or restart) a car that gives evidence of overheating.