If a paper has 3 weak accepts (in all evaluations, a weak accept in each evaluation), will it be published in the conference? If this is the case, what is the difference if it has 2 weak accepts and a strong accept or 3 strong accepts?
This largely depends on the conference. The way that conferences usually work (at least in my field for computer science - subdiscipline software engineering or human-computer interaction) is that once the reviews come in, the program committee meets and then discusses which papers will be accepted and which ones are rejected. This often comes down to how many papers there are, how many they can accept (subject to things including a mandatory acceptance rate, how big the conference can be, etc) and what other papers are like (if your conference has very few "strong accepts" that year then weak accept is pretty good).
Usually, a paper with few "strong accepts" could end up around the "middle" of the pack are more likely to be held up for discussion where they have a chance for rejection. Most PCs won't extensively discuss a paper where all of the reviewers say "strong accept".
So, we don't know if it'll be published in the conference. However, in general, the more strong accepts a paper gets, the more likely the paper will be accepted.
Note: My answer is computer science and specifically HCI relevant.
This depends on the conference, the discipline and the conference committee. There are variables which determine the acceptance of a paper in a conference.
A paper with three weak accepts may be rejected if there are stronger papers out there or if the committee feels that another paper which describes something really new must be given an opportunity to be presented.
A paper with three strong accepts is most likely to be accepted because this means that there is an almost unanimous consent among the reviewers and the paper is most likely very strong and very relevant to the subject matter.
A paper with any such combinations in between the previously described examples may or may not be accepted depending on various factors. Ultimately, the decision is upto the organizing committee. For instance, I have seen papers in CHI be rejected finally with three 3.0 scores (borderline accepts) but seen papers with one 4.0, one 3.0 and one 2.0 get accepted finally after the revisions were made.
Therefore, ultimately, my first sentence remains valid. It depends on a bunch of factors most of which are out of your direct control. The only thing which you can control is the revision of the paper.