The procedure is detailed on the IND website. I will try to clarify some misunderstandings while answering your questions but you should refer back to this procedure to know what to do.
No, you should get a Schengen visa, free of charge (the Dutch call that a “facilitation visa”). That's because you are covered by the EU freedom of movement and regular rules around immigration to the Netherlands do not apply to you. Evidence requirements to be issued this facilitation visa should be much lighter than any type D visa (including student visas) and it should be faster as well.
However, if you need to get some documents translated and/or apostilled once you reach the Netherlands (I am thinking about birth certificates for both of you and a proof of your marriage), you might also want to look into that before leaving India.
It's kind of begging the question but what you need is an address that can be used to register yourself (inschrijven) in the BRP, otherwise dealing with Dutch officialdom will be a pain. Renting a room is in principle fine as is a signed document from the main occupant of the house where you are living without a formal rental contract (I have used either of them in the past).
On the other hand, I have come across landlords who pretended that registering was not possible at their address (possibly because they violated some rules around subletting or maximum occupancy or some such) and this could create some issues. So there is no hard and fast rule that you need an independent house and a rental contract but you will need some sort of documentation and not every short-term accommodation works for that.
Why would your wife need to register with the KvK? Does she intend to be self-employed? She could do that if she needs to but it is not necessary to secure a residence card. However, your questions are relevant to the next step and I will answer them below.
Yes, indeed, that's the important step to secure your residence card. For that, the EU citizen (your wife) needs to establish either that you are financially self-sufficient or that she has a job. Here are the relevant form and a description of the full procedure.
If you go the “self-sufficient” route, both of you as a couple need to show that you have enough money to avoid being a burden for the social system in the Netherlands (i.e. you do not qualify for welfare/bijstand there). I don't know much about how this is implemented but since there is a €12450 maximum wealth threshold for welfare (vermogenstoets), having more than that on your bank account ought to be enough. You also need to prove that you have health insurance that satisfies the Dutch requirements (I wrote “both of you as a couple” because the monetary thresholds are higher than if she was alone and both need to be insured).
If your wife goes the work route, the standard to meet is that the work must be “genuine and effective”. In other words, there are no minimum income or qualification requirements. Working in a café is definitely fine, short-term or part-time work can also be enough. In that case, your finances, health insurance situation or household composition are irrelevant, even applying for public funds would be allowed if you still qualify despite having a job.
Being the spouse of a qualifying EU citizen gives you the right to be in the country (even without a visa, after crossing the border irregularly or otherwise failing to do anything). I strongly recommend following the procedure and completing all the required formalities in time but I mention this to explain that your right to be there with your wife (if she does qualify) does not depend on any visa.
In practice, the letter you get from the IND acknowledging you have applied to have your status verified should be enough to avoid trouble while the verification is ongoing. Note that the form you have to fill in is called “Application for Verification against EU law” to reflect this legal reality: you're not applying for permission, you're just applying for them to check and confirm your pre-existing right.
No, you just need one residence document (verblijsdocument). English translations are somewhat imprecise and the IND sometimes refers to the document they give to the members of an EU citizen's family as “residence permit” but this is really the same as a “residence card” (the more proper terminology in English). The IND doesn't send those through the mail, you will get a letter or notification of some sort when it's ready and you have to go to a website to make an appointment to collect the card in person. Recent experience suggests appointments are much easier to get than usual (possibly because fewer people relocate at the moment), in Amsterdam next-day appointments are available these days.
Regarding the outcome, I would not say that the decision “could go either way”. Unlike many visa decisions, there is no discretion for the authorities and no evaluation against a set of job-market criteria or anything like that. If you wife gets a job, you qualify. If she doesn't, she will have to check the other box on the form and the IND will apply the criteria for self-sufficient people. If your income or wealth are above the threshold, they have to give you a residence card. If it isn't, they will refuse. I don't know the exact threshold but it's not extremely high and there is little chance involved in all this. It's only if you have absolutely no income and no money and try to qualify as a jobseeker or based on minimal work (self-employed or employed for a few hours a month) that you would be in a grey area (technically looking for work makes an EU citizen a worker under EU law but that cannot last forever and is inherently more difficult to document than a work contract).
Consequently, I would not recommend the student visa. It can take ages, has completely different and no less onerous requirements and would give you a weaker status legally. Questions could also be raised about your intent to actually pursue studies in the Netherlands, especially if you already have a career and want to move with your wife.