The three questions you list -- "what is the purpose of life?", "do we have free will?", "do good and evil exists?" -- are ones many people have strong feelings about, and likely they do not all agree with one another.
But common sense is generally understood to mean something like "sound and prudent judgement based on a simple perception of the situation or facts" (Merriam Webster). I.e., a piece of common sense is something which is obvious to anyone and which no reasonable ("sound and prudent") person would disagree with.
This is important because it means common sense things are generally not contentious; they are obvious based on "a simple perception of the situation or facts". Citing common sense normatively means pointing out the obvious to someone, often simply by reminding them. For example, if you are in a cabin in the woods at night during a freezing cold winter, common sense dictates the door should not be left open. No one in the cabin is going to take issue with that -- unless there is another common sense factor that supplants it, for example, that the cabin has filled with smoke.
So questions which have common sense answers are generally not topics for philosophy, although the concept of "common sense" might be (e.g, contemplating how people come to recognize common sense). There are some grey areas, however:
Many common sense issues are cultural, and so come into conflict. This may reveal that they have stretched the definition of being derived from "a simple perception of [...] facts", and may involve some ingrained prejudice, which brings us to the next point:
"Common sense" is often used as a rhetorical ploy in public discourse. For example, in politics it is not unusual for a candidate to state that something his/her opponent did "defies common sense". In this case, it might be something 30 or 40 or 60 or 70 percent of a relevant population agree with, but it is still a hyperbolic use of the term, since it is meant to imply this is something "no one in their right mind" (sound and prudent) would deny. The fact that a corresponding 30 or 50 or 70 percent do deny it tells us the issue is, in fact, not one that can be decided using common sense -- and thus might be fit for philosophy.
Going back to your three questions, at least two of them are obviously not answerable with common sense unless one uses the term pejoratively (as in political rhetoric). Different people will give very different answers to, "What is the purpose of life?", and, to a lesser extent, to "Do good and evil exist?" It is also probably a matter of common sense (obvious to everyone) that the very nature of these questions makes them bad topics for common sense to tackle (since common sense requires a "simple perception of facts" and there are few simply perceived facts here).
Your approach to the third question is actually an example of philosophy, albeit with a little bit of the aforementioned political hyperbole thrown in. Is it common sense that we can make choices? Probably it is, but this is a case where philosophy might take issue, and you do, by saying
these choices depend upon many factors both conscious (e.g. someone compelling us) and not conscious (e.g. our education)
Unlike the ability to make choices, which is obvious from "a simple perception of facts",1 this is not, as you claim, a piece of common sense even if most people agree. Going back to our politician, winning an election with 70% of the vote is an accomplishment, but it does not mean the politician's hyperbole is true. That would be a perverse interpretation of democracy; it implies that if a majority vote that the sky is bright orange at night, then this means the sky is bright orange at night. Obviously a rational majority would not do that, but that is not why it is not true, either. It's not true because this is a matter of common sense, a "simple perception of facts", and not something that can be decided by human beings making choices.
Likewise, although settling debates democratically is a laudable social practice, this is not ideally how philosophy works. Philosophical issues are not published as multiple choice questions then distributed to university faculties for their members to vote on. However, there is a democratic factor in so far as in any other academic field, if your peers take you seriously, your philosophical work may flourish, otherwise it likely will not. But the argument that something is false because it is unpopular is not a good one; ideally, we want a rational case for the opposite: that it is unpopular because it is false. This does not free us from making a case about why it is false (again, not because it is unpopular).
Framing an issue as a matter of common sense is political rhetoric if the issue is in fact not one which everyone will easily agree upon, and this is exactly what you have done with the issue of free will; despite the fact that almost everyone will agree that for pragmatic purposes we can make choices, many of them will admit that could be a bit of a theoretical distinction for various reasons. So free will is a good topic for philosophy precisely because it turns out to be a bad topic for common sense.
To answer your question, then, no, philosophy does not work by tackling common sense questions and "adding complexity but not answering the question more precisely". Truly common sense questions ("Should I leave the door open?") are not really serious topics for philosophy. We might even say that philosophy begins where common sense ends, and this is true to the literal meaning, "love of wisdom or knowledge", if we assert that the "wisdom or knowledge" referred to here is potentially of the uncommon sort.
1. This is not to say that strong determinism is wrong, only that it defies common sense. Just because common sense is defined as something which is obvious to everyone does not mean it is always correct (although it is probably political suicide to challenge). There are often cases where what was obvious to everyone turns out to be wrong; a flat earth is a prime example.