You read the dates and convert them into the class datetime. For this you need to read about datetime, datenum and datestr. Once you have dates and data in hand, you can play with the indices and get the data you want.
I will make the datetime data and answer your questions.
Answer 1:
thedates = [datetime(2022,1,1):datetime(2022,12,31)]' ; % this is the datetime data, you need to get from nc files
class(thedates)
% pick each 10 days data for Jan
idx = thedates.Month==1 ;
t = thedates(idx) ; % dates for Jan month alone
idx1 = t.Day<=10 ;
t(idx1)
idx2 = t.Day > 10 & t.Day <=20 ;
t(idx2)
Answer 2:
idx3 = thedates.Day == 1 ;
thedates(idx3)
Answer 3:
idx4 = thedates.Month == 3 & thedates.Day == 4 ;
thedates(idx4)
Like above, you can get the indices. The same indices should be used to extract the data.
