"It is the mark of an educated mind to be able to entertain a thought without accepting it." --- Aristotle.
There seem to be professors who hold beliefs such as [...list of beliefs I don't like]. No student should be saddled with negative prejudices from their supervisor. But how can this be prevented?
One of the things you should be very careful of, especially when you are beginning your tertiary education, is to avoid coming into that educational environment with views that are so rigid that they leave no room for consideration of theories and evidence pertaining to those ideas. In your question you set out a list of beliefs you don't like (many of which are empirical claims, and some of which are almost certainly true) and you then blithely declare that all these claims are mere "negative prejudices" held by "bigots". At a holistic level, your question is essentially asking how you can go through your higher-education without having to encounter beliefs that you don't like (on the purported basis that people holding those beliefs might harm you or others in some way).
This is going to cause problems for you in terms of seeking an education to train you to think more clearly, and to be able to entertain ideas with which you disagree. In each of the cases you list, there is a claim made, which can be analysed by applicable theory and empirical evidence. Unfortunately, you don't seem to be at all interested in doing that, and you are instead seeking advice on how to prevent these ideas being believed at all. At best, this attitude is likely to hinder your ability to learn how to analyse a claim for something you disagree with, and you will need to work hard to be able to approach these claims with an analytical mindset. At worst, it will make you uneducable --- you will attend an institution of higher learning, but your rigid attitude will prevent you from engaging with contrary ideas in an analytical way.
- Women students pose a special risk to male supervisors
Whether or not this claim is true really depends on what you mean by a "special risk". It is unlikely that there is a distinct category of risk that applies only in this particular case, but it is at least arguable that there are some risks that are magnified when a male supervisor supervises a female student. As you are no doubt aware, most people in society are heterosexual, and male heterosexuals are empirically more likely than female heterosexuals to engage in sexual harassment of subordinates, and their behaviours are probably also more likely to be perceived as harassment under similar circumstances. (I could be wrong about this, but it is at least arguable on good faith.) Thus, it is certainly arguable that, ceteris paribus, the risk of sexual harassment increases when a male supervisor supervises a female student, as does the corresponding risk of a false claim. Ultimately, a claim of higher/special risk needs to be assessed against empirical evidence of outcomes in different supervisory relationships. It is foolish to rule this out as a mere "prejudice" in the absence of some attempt to engage the claim.
- Older students are worse investments than younger students
There is a very strong argument that this claim is true. If students are to be considered as an "investment" at all, then that must presumably mean that they will generate some future benefits (e.g., for society) and their "value" as an "investment" depends on the frequency and magnitude of those future benefits. In economic theory, the value of an investment is determined by some calculation of the "expected net present-value" of the stream of future benefits from the investment. Ceteris paribus, a student with more remaining life is likely to give a longer stream of future benefits, and is thus a higher-value "investment" than a student who has less remaining life. It follows that, ceteris paribus, a younger student is a more valuable "investment" than an older student operating at the same level.
I'm sure there are probably some reasonable arguments against this view, but the point is that you would need to actually listen to these competing arguments and evaluate their merits to decide on the truth of the claim. In view of the strength of the supporting argument for this claim (it is virtually a logical consequence of the economic definition of expected present-value, plus the fact that younger people have higher remaining life-expectancy), it is hard to see how this claim can reasonably be characterised as a "prejudice" held only by "bigots".
- Certain ethnic groups are smarter/lazier/harder-working than others
This is an empirical claim, and the only way to resolve it in a robust way would be to look at data on those characteristics (measured somehow) for a range of people in different ethnic groups. There is a mountain of empirical literature on IQ, etc., and this literature consistently shows differences among race/ethnic groups. Studies on levels of leisure-time and physical inactivity give more mixed evidence, but there are usually differences among race groups found in individual studies. Whether or not those various differences are "nature or nurture", and whether or not they are likely to remain over time, etc., are controversial topics on which much academic literature has been written.
There are probably all sorts of nuanced and reasonable scientific views one could hold on this topic. Many academics argue either that the measures of intelligence, etc., are flawed, or that the empirical results of difference groups are likely to converge over time, or make other claims. Others do indeed hold that there are differences in these characteristics rooted in genetic causes. The particular claim you mention is a pure empirical claim (i.e., a descriptive non-causal claim) and can be assessed directly by data on the present population. In view of the fact that this is an empirical claim, beliefs about the claim by academics are likely to be influenced by empirical data, or at least anecdotal observation, and it is thus unlikely that the claim would mere be a "prejudice". If you have a look at the academic literature on race/ethnic differences in various positive characteristics like intelligence, diligence, etc., you will see that there are a range of views and a lot of study and data that can be used to elucidate the topic.
- Students with disabilities are less capable/more trouble than nondisabled students
A dis-ability is literally an incapacity --- i.e., something that makes someone less capable of doing something. Thus, the claim that a person with a disability is less capable (somehow) is a tautology, taking the concept of "disability" and replacing it with its synonym "incapacity". Now, obviously any given disability gives a particular set of incapacities, and those limited incapacities do not necessarily imply any broader lack of capability beyond the specifics of the disability. As to what particular incapacities or "trouble" that would imply, it would depend on the nature of the disability.
It should go without saying that a university should try its best to accommodate student disabilities, and supervisors should be willing to go to some "trouble" to assist people with disabilities to the extent that this is reasonably required. Nevertheless, with respect, treating the above claim as a "prejudice" held only by "bigots" is really quite idiotic. It is a claim that is virtually a tautology, and any denial of the claim would render the concept of "disability" meaningless.