Is my approach to discouraging whiners from taking my class acceptable?
Frankly, the whole thing seems like a terrible idea - a guaranteed way to sabotage your rapport with your students and ensure complaints on your evaluations. If you're trying to get tenure, this seems like a good way to make sure you don't get it.
I suspect that the "if you don't like it, withdraw" clauses won't have the desired effect of weeding out all the problem students. Rather, they'll weed out everyone except those who have no choice but to take the class (required for their major, can't wait another semester, other sections don't fit their schedule). Neither you nor the remaining desperate students are going to be happy to be stuck with each other.
But I'll comment on #5 specifically. I believe this is unethical, and probably contravenes your university's disciplinary rules. While obviously there is wide variation in how cheating cases are handled, I would say that a common principle is that the student has "due process" rights. You can't unilaterally impose a punishment (and I would say that a more difficult exam is certainly a punishment) based solely on your suspicions. The student has a right to see the evidence against them, defend themselves, and appeal to a higher authority.
I see a basic disconnect between the course title "Calculus for Business" and the stated objective "The main goal is to help you learn how to teach yourself math. I will not be able to teach you all the math you need to know, either for this course or for the rest of your life. You will have to teach yourself some of the material by reading the textbook. If you can't learn by reading a textbook, please withdraw."
I would expect a course called "Calculus for Business" to be suitable for business administration and similar students who need to learn some calculus for finance and statistics, and as part of their general education. They are outside their comfort zone, trying to learn a subject for which some of them are not especially talented, and that many of them do not enjoy. Some of your students may need both a lecture and another explanation in a textbook to get a concept. This is likely to be the last, and most advanced, mathematics course they ever take.
On the other hand, the goal would be suitable for a course targeting second or third year mathematics majors who have a lot more mathematics to learn and already have significant exposure to college level mathematics textbooks.
The fact that you cannot teach, in your lectures, all the material you expect them to learn is another indication that you are asking a bit too much for the population and the length of the course.
An inappropriate goal for the student population could be the root cause of a lot of friction.
I suggest consulting with your department to make sure your objectives and the demands you are putting on the students match the department's objectives for the course and the target student population. You may need either a different course title to make the nature of the course clear, or a revised syllabus.
I see several issues with your syllabus draft that are worth thinking about some more.
Course objectives, goals, policies, etc:
1) The main goal is to help you learn how to teach yourself math. I will not be able to teach you all the math you need to know, either for this course or for the rest of your life. You will have to teach yourself some of the material by reading the textbook. If you can't learn by reading a textbook, please withdraw.
Your “main goal” is very commendable, but it is hopelessly unattainable within the scope of a calculus class. The skill of being able to teach oneself math from a textbook takes several years to master, and I would not assume a student has it (or is capable of acquiring it) until they are at an advanced stage in a math graduate program. So the last sentence is effectively telling 99% of your students they need to withdraw from the class.
2) Whining will not be tolerated. If my course is too demanding, please withdraw.
I second the advice in @mathochist’s answer not to use the word “whine” or any of its derivatives. Moreover, I would wonder why you even think it makes sense to have the sentiment that whining should “not be tolerated”? Your job includes dealing with students who will sometimes struggle with your class and may give voice to their unhappiness; this is simply human nature. So, tolerating “whining” is in your job description.
Instead of putting draconian restrictions on your students about what they may or may not say, perhaps you can work on your own attitude, and give some thought to how to teach in a way that leads students to be happy instead of feeling like whining, or, if they do whine, how to deal with that effectively or even turn their whining into a teachable moment with some positive benefit? Just a thought.
3) The goal is for you to do mathematics, not just to``know'' it. The more math problems you do by yourself, the better. Math eventually becomes fun when you do things yourself.
There is a wonderful rule in writing and communication called “show, don’t tell”. You will never convince anyone that math is fun by saying things like “math is fun!” Instead, you have to show them that math is fun. So, while the sentiment you are expressing here is very valid and laudable, this paragraph serves no useful purpose other than to clutter up your syllabus. I suggest removing it.
4) The purpose of lectures is to help you keep pace with the material, and show some of what you need to learn. (And to address math questions you had). The purpose of lectures is not mainly to teach you the material, although there will of course be some of that.
I also don’t understand what purpose this paragraph serves. I suggest keeping you thoughts about what the purpose of lectures is to yourself. Lectures are a standard part of university life and everyone has their own idea about what they’re good for and how useful they are (with some students considering them very important, others less so). Do you really think any one of the small number of students who bother to read this part of your syllabus will change their behavior in any way, or stop and say to themselves “wow, I never thought of it that way! This has completely transformed the way I think about lectures”? I don’t think that’s very likely...
5) Students who I suspect have cheated, whine about how difficult the course is, or are disruptive during class may get slightly more difficult exams than other students. If you have a problem with this policy, please withdraw from this class.
This is just awful, for reasons explained well in the other answers.
To summarize, it sounds like you have noble ideas about teaching math, but at the end of the day, you are teaching calculus, not philosophy. I would reserve the syllabus for listing the actual topics you plan to cover and other practical matters like a grading policy etc (you’d be lucky if most students even read those parts), and leave the philosophical discussions to those few minutes here and there during lecture if/when it makes sense or you are in a particularly philosophical mood. And please try to be more tolerant of students’ venting about their struggles in your class.