How can I “merge” rows by same value in a column in Pandas with aggregation functions?

You are looking for

aggregation_functions = {'price': 'sum', 'amount': 'sum', 'name': 'first'}
df_new = df.groupby(df['id']).aggregate(aggregation_functions)

which gives

    price     name  amount
id                        
1     130     anna       3
2      42      bob      30
3       3  charlie     110