a) the minimal points possible to advance: 2 (!)
If one team beats all the others and the other three all draw with each other, points are distributed the following: 9-2-2-2 

b) the maximum points possible to be sent home with: 6 (!)
If one team loses all matches and the other three beat each other the following: A beats B, B beats C, C beats A, points are distributed the following: 6-6-6-0 
remark: this is a similar case to the one, Algeria suffered from at the 1982 World Cup. 

c) going home undefeated: 
1) one team loses all, matches, the other three draw with each other. Point distribution: 5-5-5-0. 
2) two or four teams end up with 3 draws each. Point distribution: 5-3-3-2 (Cameroon at the World Cup 1982 were the victims), or 3-3-3-3 (all group matches end with a draw). 
3) one team ends up with 3 draws but none of the other matches end with a draw: Point distribution: 7-4-3-1 or 4-4-4-3 
4) one team (A) ends up with 3 draws, B and C draw and both beat D: Point distribution: 5-5-3-1 

d) advancing without winning a match: 
1) one team wins all, matches, the other three draw with each other. Point distribution: 9-2-2-2. 
2) two or four teams end up with 3 draws each. Point distribution: 5-3-3-2, or 3-3-3-3 (all group matches end with a draw). 
3) one team (A) ends up with 3 draws, B and C draw and both lose to D: Point distribution: 7-3-2-2 

e) losing two matches and advancing anyway: 
One team wins all three matches and the other beat each other A beats B, B beats C, C beats A. Point distribution: 9-3-3-3 

f) If there is a draw in at least one of the four matches of the first two days in the group: 
At least on one of the two teams to advance will not be decided upon before the third match. 
Even for a team with only one draw and a loss there will be always a theoretical szenario of advancing before the last match. 
A team that has not won both matches, cannot be qualified for the next round before the third match. 

g) The most boring case: 
Two teams have 6 points, the other two 0 points after the first two matches. The final match only decides on first and second place of the group. (France 98, group H with Argentinia and Croatia). 

h) After two matches: A team can be eliminated when losing the first two matches, depending on the outcome of the others.
If at least one of the other two other matches has ended with a draw, they will have to go home. 
Otherwise it depends on who has beaten whom. 

i) After two matches: A team can have reached second round winning the first two matches, depending on the outcome of the others. 
If at least one of the other two other matches has ended with a draw, they are through. 
Otherwise it depends on who has beaten whom. 

j) A problem case: 
One teams is already 1st place for sure after two matches (a 3-point-rule implication). They have nothing to play for (in opposition to a team that is out after 2 matches who at least play for pride). It happens when one team wins both matches and the other two matches end in a draw. Point distribution: 6-2-1-1. Morocco and Spain at France 98 can tell you a story here: Brazil and Nigeria lost their last matches in rather lacklustre performances against Norway and Paraguay. 
France and Portugal in Euro 2000 fielded a second string team. What at first triggered protests from the other teams turned out to be the right choice: for the reserve it was 'their' big game. Playing with 100% commitment they defeated their opponents.

k) The critical 3rd match

 The African Cup Of Nations 1998 shown: There is a special critical third match situation. 
 No group had been decided before the last match. This last match such became a kind of play-off match, but with a different, tricky situation: Some teams had to go into the matches, defending their positions. They needed only to keep the score even, achieve a draw against opponents who wanted to win. This played a role in succession of the matches when those favorites were pushed more and more backwards and in two matches a late goal conceded left them no time to strike back. Ghana and Guinea were eliminated because of this.
 Recent tournaments have shown that teams who could approach the third match relaxed, already qualified, seemed to benfit from this later. You might argue with this interpretation: The team might have just been so strong.
 Another point is the succesiion in which the teams play the prime team, of the group. This often gives an edge to the team who plays this prime team third. Think of Norway and Paraguay in Frnce 98 who advanced not at last thanks to playing Brazil and Nigeria when those already were qualified. 

l) The critical 1st match

 No World Champions have lost their opening match.