I can go into a little bit..
First off, great question. Almost every single one of us in the internal arts has asked similar questions, especially because it is so often subjective.
I will not go into too much detail with your questions, but I will answer a few of them with some brief things to pay attention to:
“What’re the characteristics I should watch for in the class and the teacher, that might reassure me that they’re genuine in their fighting ability?”
- What do the more advanced students have to say? Does the teacher emphasizes the fighting aspects and health aspects equally? Are you taught any martial aspects early on in the beginner classes such as how to do a taiji punch?
Some things to watch for in the teacher: Do they sink into their moves, for instance when stepping does it appear the body sinks into each move so it can be executed slowly and smoothly?
Do they appear completely relaxed? If so, this would be a good starting sign.
Can they perform anything they say they can? If they say they can do something, call them on it, make them show you. If they don’t then they are hiding it for some reason. Ask them what reason.
But my favorite way to tell if a taijiquan instructor knows how to fight is to spontaneously attack them. It gets you thrown down usually if you do it, and a little bruised, but its usually a great way to find out. You have to work up some nerve to do this one, but if your teacher is experienced, they probably wont hurt you. 
What type of drills should I expect? What about approaches to training?
You should expect basic qi gong (hugging the tree, wuji, maybe a few others), and the first parts of the form they decide to show you. Also a good teacher will emphasize how the entire body should be used for each move you make, like how the waist must turn while punching etc. They probably won’t do much talking about “qi” and mostly refer to the breath. If they talk endlessly about qi, you might ask them why they talk about it so much or tell them its confusing you.
Can any of you detail your own growth in the art, from the beginning to a level of actual combat skill?
I think this will 100% vary from one person to the next. I know people who have spent 2 years learning the 24-movement form and people who picked it up in 5 months and then went on to the YCF (Yang Cheng-Fu) old style. Some people take 6 years to learn that, some people take 1-2 years. It depends on your perserverance, will power, willingness to learn, how often you practice, discipline etc.
You probably won’t be able to fight effectively using taijiquan in the first two years, but I would say usually in 3-5 years you should be able to express some martial abilities. In about 10 years you should definetely be able to fight well. This is of course dependant on if you train properly and how often your train.
Freedom is what you do with what is done to you. - Sartres