Pandas: How do I assign values based on multiple conditions for existing columns?

You can do this using np.where, the conditions use bitwise & and | for and and or with parentheses around the multiple conditions due to operator precedence. So where the condition is true 5 is returned and 0 otherwise:

In [29]:
df['points'] = np.where( ( (df['gender'] == 'male') & (df['pet1'] == df['pet2'] ) ) | ( (df['gender'] == 'female') & (df['pet1'].isin(['cat','dog'] ) ) ), 5, 0)
df

Out[29]:
     gender      pet1      pet2  points
0      male       dog       dog       5
1      male       cat       cat       5
2      male       dog       cat       0
3    female       cat  squirrel       5
4    female       dog       dog       5
5    female  squirrel       cat       0
6  squirrel       dog       cat       0

using apply.

def f(x):
  if x['gender'] == 'male' and x['pet1'] == x['pet2']: return 5
  elif x['gender'] == 'female' and (x['pet1'] == 'cat' or x['pet1'] == 'dog'): return 5
  else: return 0

data['points'] = data.apply(f, axis=1)

numpy.select

This is a perfect case for np.select where we can create a column based on multiple conditions and it's a readable method when there are more conditions:

conditions = [
    df['gender'].eq('male') & df['pet1'].eq(df['pet2']),
    df['gender'].eq('female') & df['pet1'].isin(['cat', 'dog'])
]

choices = [5,5]

df['points'] = np.select(conditions, choices, default=0)

print(df)
     gender      pet1      pet2  points
0      male       dog       dog       5
1      male       cat       cat       5
2      male       dog       cat       0
3    female       cat  squirrel       5
4    female       dog       dog       5
5    female  squirrel       cat       0
6  squirrel       dog       cat       0