To answer your last question first, you can check out the monthly Equatorial Southern Oscillation numbers here or any other numer of indices here. 

As for the SST indices, we use the ERSSTv4 for verification and classification purposes (and you can find it in the list of indices at the second link above) which come out once a month. Other SST datasets like the OISST come out every week in addition to monthly and seasonal values. Both use different processes to determine sea surface temperatures which accounts for their differences in Nino3.4 values.  ERSSTv4 has a longer record which allows for better comparison to previous events. OISST only goes back to 1981 while ERSSTv4 goes back to 1950. 

If you average the monthly ERSSTv4 values, they will equal the seasonal 3-month averages as well.