When I applied for the visa appointment, I had entered "tourist visa". However, I later came to learn that I would need a business visa. Will that be a problem?
No, it doesn't matter much as long as you do not give the impression that you are trying to disguise the purpose of your trip. There is no such thing as a business or tourism visa in the Schengen area but only one kind of visa for all stays of 90 days or less (“short stays”). “Business” and “tourism” are only two of nine potential purposes listed on the standard application form and it's also possible to pick several of these purposes or check “other”.
What does matter is being upfront about the purpose of your trip and providing a strong documentation that lines up with this purpose. Since the conference is the main reason for the trip, you should mention it everywhere you can (including on the form) and provide relevant documentation to support your application (conference invitation, bookings for your stay near the conference venue, proof of employment at the institution paying for your trip if that's relevant).
I plan to travel a couple of weeks early for the purposes of tourism. Is that allowed on a business visa?
Your visa will in principle be valid for all purposes. What's tricky is using a visa for a different place, time, or purpose than originally intended. As mentioned earlier, it's perfectly fine to have several purposes for a trip but makes sure to provide a detailed itinerary for the tourism part and include as much evidence as possible for that part of the trip as well (things like hotel bookings, etc.)
Will be a issued a three-month visa such that I can enter and leave whenever I want within those three months?
No, you cannot count on that, especially if this is your first visa application. Your visa is likely to be limited in duration of stay, dates of validity and number of entries. That's another reason why it's important to be upfront about the purpose of your trip and explain what you plan to do in detail. Again, what you select when booking the appointment is not hugely important but the contents of your application do matter.
PS: I am assuming that you do need a visa and have addressed your question as asked but as noted in the comments, not everybody needs a visa to travel to the Schengen area for this purpose. Conferences will provide information and invitations for everybody who needs a visa but if you are a citizen of a country listed in annex II of this document you can safely ignore all that.
