How to drop multiple rows with certain 1st level and 2nd level index?

While it is typically recommended that each StackOverflow question should be limited to an single issue, these are close enough to being the same, that I will provide my solution for doing what you are looking for:

Given a df like:

                A   B
Name    Date        
AA  2018-01-31  -1  52
BB  2018-02-28  0   94
CC  2018-03-31  6   86
DD  2018-04-30  3   50
EE  2018-05-31  11  60
FF  2018-06-30  9   117
GG  2018-07-31  0   45
HH  2018-08-31  -3  62  

# Drop a single row
df.drop('AA', level=0, axis=0, inplace=True)  

Which removes the Name 'AA' from the dataframe and will in fact remove all 'AA' indexed items

To remove multiple rows you can use:

# Drop several timestamps

    df.drop([pd.to_datetime('2018 03 31').date(), pd.to_datetime('2018 07 31').date()], level=1, axis=0, inplace=True)  

In the case where you have multiple items indexed at level 0 but you want to remove one or more items from level 2 index you can use the following:

df.drop(('CC', pd.to_datetime('2018 03 31').date()), axis=0, inplace=True)

I m going to provide an answer based on the following dataframe example (you had to provide one actually):

                          columnA columnB
NameA 2016-01-01 12:00:00       p       a
      2017-01-01 12:00:00       q       b
NameB 2018-01-01 12:00:00       r       c
NameC 2019-01-01 12:00:00       s       d

how can I drop one row with a certain 'name' and 'timestamp' index?

Lets say you want to drop name with 'NameA' and timestamp with '2017-01-01 12:00:00' then you could use:

df.drop(('NameA', pd.Timestamp(2017, 1, 1, 12)), axis=0)

output:

                          columnA columnB
NameA 2016-01-01 12:00:00       p       a
NameB 2018-01-01 12:00:00       r       c
NameC 2019-01-01 12:00:00       s       d

how can I drop multiple rows for one name and a list of timestamps?

You can use pd.MultiIndex.from_product to create a multiindex that you want to drop.

Example: you want to drop the two timestamps that are for 'NameA':

df.drop(
    pd.MultiIndex.from_product([
        ['NameA'], 
        [pd.Timestamp(2016, 1, 1, 12), pd.Timestamp(2017, 1, 1, 12)]]),
    axis=0
)

output:

                          columnA columnB
NameB 2018-01-01 12:00:00       r       c
NameC 2019-01-01 12:00:00       s       d

for dropping multiple rows for one name and a list of timestamps, you can take help of utility function like

def drop_multi(df, ind_level_0:str, ind_level_1:list):
   for ind_1 in  ind_level_1:
      df.drop((ind_level_0, ind_1), axis=0, inplace=True)

then call this function with desired arguments, in your case -

drop_multi(df,'Name', list_of_timestamps)