Look in the cumulative binomial probability table with n=8 
and look at the column with p=.5.

You can find such a table here:

http://www.statisticshowto.com/tables/binomial-distribution-table/

You will see this.  The values are really p(x ≦ the value listed)
although the heading just reads "x": 
   
x≦  p=.5
-------
 0  .004
 1  .035
 2  .145
 3  .363
 4  .637
 5  .855
 6  .965
 7  .996
 8    1 

1. Find the probability that your sample has exactly 6 women. 

p(x=6) = p(x≦6) - p(x≦5) = .965 - .855 = .110

2. Find the probability that you have at least 6 women.

p(x≧6) = p(x≦8) - p(x≦5) = 1 - .855 = .145

[Notice that that is the same as p(x≦2) because if there are at least
6 women there are 2 or fewer men, and the probability of a man and a 
woman are the same.]
 
3. What is the probability that there are at most 0 women.

That's the same as p≦0 = .004

4. Find the probability that there at least as many men as there are women

That's the probability that there are 4 or fewer women.

p(x≦4) = .637

Edwin