Hello HappyAccidental,
Haven't been on here in a while, but saw your post and thought I'd reply.
First of all, you made the correct decision with the 9-year-old. You are right that perfection is not the goal. But you still should have a goal, and consider how, in the long-term, your student might progress towards and past Dad's standard for performance. It would help more to hear what the piece was and how you and Dad evaluated the performance differently. Also, it might help to think about why your student has a difficult personality. Unfortunately, with challenges like ADHD and Autism, parents may not tell the piano teacher or even know themselves, and it's not the piano teacher's place to come forward about it. I've also heard that piano lessons are often where dyslexia is discovered, though I haven't had a case myself. Try reading up a little on these challenges and see if there is anything that might help you restructure the lesson so the child learns more smoothly.
Let me look at your questions one by one, and I hope other teachers will reply too:
Is there a best methodology to assess a new student who's about to start with you?
I find this is a tough question to answer. The reality is, there's only one first lesson with a new student, and half an hour isn't exactly a lot of time. When I have a student who has studied before, what I have in mind are the elements of music. I know that I prefer to work from the Music Tree series by Clark, Goss, and Holland, so I have in the back of my head the progression of musical elements in these books and where the student might fit in, in terms of level. Of course, students who aren't beginners have been taught different things in different orders, so you'll have to consider if an element can be picked up as you go, or if you'll need to assign a piece or two from an earlier book specifically addressing that element.
As far as if you're looking to see if a student is ready, you'll generally be able to tell. Students who aren't ready will take every chance they get to sidetrack you. They may simply not do what you've asked and tell you a story about what they did in school that week. Of course, you politely listen, laugh, and engage them, but in the end, they must be open to your suggestions and learning. Always be kind, but it isn't wrong to say to a parent, "I think it might be best if you wait a year before trying piano lessons."
What method books best apply to which type of student?
This is an approach to method books I generally do not agree with. The reason being that many methods are simply poorly designed. However, you must understand that I received my pedagogy training at Westminster Choir College, a program founded by Frances Clark, one of the greatest researchers in pedagogy. So, when I look at the Music Tree series, everything makes a beautiful sort of sense, and when I look at other methods, I'm not always confident the authors were thinking in how they structured the books. I can tell you Bastien is far outdated and will never be appropriate. The new Alfred Premier Piano Course seems to have some great ideas, and was written by some of the best pedagogical composers of our time, but I have not tried it yet. It is something to study and try out. Faber and Faber generally works if you do your best to teach as if it were the Music Tree. I think the actual advice in Piano Adventures on reading is horrible, but the Technique and Artistry books are great, I really like how the Theory books provide ear and eye training activities, and the repertoire is very catchy and exciting. I only really get queasy because the method of learning notes doesn't show any sort of consistency, and lacks some of the valuable features of the Music Tree.
My best advice is this: go get Questions and Answers by Frances Clark (available at
https://www.francesclarkcenter.org/shop?product_category=Books) and A Piano Teacher's Legacy by Richard Chronister, as well as a complete set of the Music Tree up through Level 2B (get complete sets of any other series you might want to study and compare as well). Teacher's guides are never a bad idea, also. In addition, read
https://www.pianopedagogy.org/practice-steps-for-successful-independent-learning/. Angela wrote such a brilliant article, and she actually went to undergrad with me (she was finishing up when I started), though she attended Oklahoma University, also one of the top piano pedagogy schools in the country. The blog is, of course, full of good information, and the New School has a history of being a center for pedagogy research (Again, also founded by Frances Clark. I'm very sorry, she's kind of an idol for me).
How do you speak with parents in such a situation as above?
Kindly, compassionately, understandingly, but most of all with authority. Unfortunately, part of the game is impressions, and you need to give the impression that you know exactly what you are doing and that is why he is paying you to teach. Even if you're afraid you're wrong, you can't talk without certainty. Maybe next time save the last five minutes for discussion with Dad. Think ahead, though. You need to explain the decisions you make and account for his concerns. Always reiterate that patience and support on his part will be the greatest assets to his child. I don't know what actually happens, since you didn't say, but if he's unhappy with you letting him go on from this piece, he may also be providing extra criticism for his son at home. If so, he needs to be reminded that his son does not need extra teaching from him. What he needs is for his father to listen as if hearing his son play the piano is the only thing that gives him pleasure in this world.
If you are having trouble feeling confident about the discussion, read this:
https://www.theguardian.com/news/oliver-burkeman-s-blog/2014/may/21/everyone-is-totally-just-winging-itHow do you best acclimate to this type of environment?
By making it into an environment. The truth is, private teaching is kind of isolated, so you need to do your best to make yourself part of it. I'm sure you're doing this already, and it's just going to take time. Ask your boss for opinions on pedagogy or managing children in ways that don't undermine your authority. Communicate with the other piano teachers, ask what methods they use, how work is going, even about their personal lives. Stop in and see the receptionist(s) and ask how he/she is doing. It's worth staying an extra few minutes after you teach to chat with people. I find I like my coworkers, and it provides the balance needed to sitting alone in the room with one child after another. Parents may be asked about their lives, too. And, if you decide to strike out on your own, go out to lunch with other piano teachers in the area. The actual teaching of the kids will get comfortable if you plan and reflect. Just make sure that you make a bigger web of connections for yourself as well.
One last thing I wanted to mention is that you should remember not to limit yourself to the method books. Try to keep all fun activities geared at the keyboard, but it is nice to try a piece or two from another method, or one of the countless pedagogical sheets out there. Two composers to start with are Dennis Alexander and Martha Mier. You will find things as simple as middle elementary. Also, teach by rote. Go Tell Aunt Rhody, Jingle Bells, etc. You can make up simple accompaniments, even just I/V or blocked fifths. This helps develop their ears further when the eyes might be taking time to catch up. I've got a starting list of rote pieces if you'd like.
I really hope this helps. Please let me know if you need anything else!