Series.str.contains has a case parameter that is True by default. Set it to False to do a case insensitive match.

df2 = df1['company_name'].str.contains("apple", na=False, case=False)

It does work for me the only problem I found was that you are selecting col1 instead of Col1. The following code:

df1 = pd.DataFrame({'Req': [1,2,3],
                    'Col1': ['Apple is a fruit',
                             'Sam is having an apple',
                             'Orange is orange']})
Apple=df1[df1.Col1.str.contains('apple',case=False)]
print(Apple)

prints

>>>    Req                    Col1
>>> 0    1        Apple is a fruit
>>> 1    2  Sam is having an apple

You could either use .str again to get access to the string methods, or (better, IMHO) use case=False to guarantee case insensitivity:

>>> df = pd.DataFrame({"body": ["ball", "red BALL", "round sphere"]})
>>> df[df["body"].str.contains("ball")]
   body
0  ball
>>> df[df["body"].str.lower().str.contains("ball")]
       body
0      ball
1  red BALL
>>> df[df["body"].str.contains("ball", case=False)]
       body
0      ball
1  red BALL
>>> df[df["body"].str.contains("ball", case=True)]
   body
0  ball

(Note that if you're going to be doing assignments, it's a better habit to use df.loc, to avoid the dreaded SettingWithCopyWarning, but if we're just selecting here it doesn't matter.)

(Note #2: guess I really didn't need to specify 'round' there..)

This should work - need to convert the index to a Series first.

df[pd.Series(df.index).str.contains('base', case=False)]