The minimum number of matches that a team who advanced to the next stage has won?

From the solution of previous question even after winning 5 matches a team can get eliminated so to be sure a team must win 6 matches.

In the 1^st round in one group number of matches is (7 x 8/2) = 2
Lets consider bottom 3 they will have 3 matches between them, so remaining 28 - 3 = 25 matches have involvement of top 5 teams, if they won equal number of matches i.e 5 each then decision will be taken based on tie breaker rule, hence a team may be eliminated even after winning 5 matches.

Number of matches in Stage 1 is 2(⁸C₂) = 2(7 x 8/2) = 56
stage 2 will be a knockout tournament with 8 teams so in this stage number of matches will be 7.
Total number of matches is 56 + 7 = 63

Minimum is 6 and maximum is 10 represented by option

No player can win 3 matches

From the solution of previous question a team even with two wins can advanced to next stage where it has to play 3 matches so total number of matches is 5.

Let total number of teams participated in tournament is n + 10
There are 10 teams in the bottom group then n teams in the top group. It is given that the bottom group gets 45 points since we have 1 point per match therefore 45 matches playing amongst themselves. Therefore they should get 45 points from their matches against the top group i.e., 45 out of the 10n points. The top group get ⁿC₂ points from the matches among themselves. They also get ( 10n ? 45 ) points against the bottom group, which is half their total points.
Hence ⁿC₂ = 10n ? 45 ? n(n + 1) = 20n ? 90 ? n² ? 21n + 90 = 0 hence n = 6 or 15
If n = 6, the top group would get ⁿC₂ + 10n ? 45 = ⁿC₂ + 10(6) ? 45 = 30 points, or an average of 5 points per team, while the bottom group would get (45 + 45)/10 = 90/10 = 9. This is not possible. Hence n = 15. Then total number of teams is 10 + 15 = 25.

1^st let us understand the concept of the tournament.
Try to understand the table, in the 1^st round the sum of seed is always 33, the same for round 2 is 17 and so on considering there is no upset.
Lindsay Davenport (Seed 2) was scheduled to to meet seed 7, i.e., Justine Henin in the quarter finals. But in the second round since match 7 - involving seed 7 (Justine Henin) and seed 10 (Venus Williams) - resulted in an upset and Venus Williams won the game, hence Davenport will meet Venus Williams in the quarter finals.

If the top eight seeds make it to quarterfinals, then from the table we can find that Sharapova being top seed will meet seed 8 in the quarterfinals, and in the semifinals she will meet the winner of the match between seeds 4 and 5. So she will not meet seeds 4, 5 or 8 in the finals.
We can sum up the given information as below:
(ii) No team plays against the same team more than once
(iv) No ties permitted.
As per the information given for stage I, we can conclude and draw the table that
(a) As B lost at least one match, hence A won all the 3 matches.
(b) The two teams who lost all the matches cannot be A (as it won all 3 matches), cannot be B (since it is given that E lost to B), cannot be D (since D won against C & F). Hence, the two teams must be C and F.
(c) F did not play against the top team (i.e. A).
We can tabulate all the information as :
In the given below table a blank space means there was no match in this stage (i.e there was no match between A and E in this stage), Result of a match for a team is shown in its row,
(d) A lost both its matches against E and F.
(e) F won against A, hence is the bottom team (out of C and F) which won both the matches F won against C as well. This also means that C lost both its matches against B and F.
(f) Apart from A and C, one more team lost both the matches in Stage-II. That team can neither be E (since A lost to E), nor B (since C lost to B), nor F (as F won both its matches). Hence, the team must be D.

All odd numbered seeds up to 15 reach the 2nd round while instead of seeds 2 to 16 the players who reach the second round are seeds 31, 29, 27, 25, 23, 21, 19 and 17.
The second round matches are between seeds 1-17, 31-15, 3-19, 29-13, 5-21, 27-11, 7-23 and 25-9. Since there are no upsets in that round the winners are seeds 1, 15, 3, 13, 5, 11, 7 and 9.
The quarter final line up would be
1-9, 15-7, 3-11 and 13-5.
If seed 13 wins her match in the quarters she will next meet seed 1, i.e. Maria Sharapova in semifinals.