I would like to know if there are online available and free world-wide datasets of sea surface temperature, net primary productivity, salinity, pH and dissolved oxygen.
there is a global data set called Bio-Oracle, which could be useful. It has the aim to be used in marine species distribution modelling, and covers the variables you listed except pH. The dataset contains both present-day and future values until the end of the Century.
http://www.bio-oracle.org/index.php
Scientific references which describe the data:
Assis, J., Tyberghein, L., Bosch, S., Verbruggen, H., Serrão, E.A. & De Clerck, O. 2018. Bio-ORACLE v2.0: Extending marine data layers for bioclimatic modelling. Glob. Ecol. Biogeogr. 27:277–84.
Tyberghein, L., Verbruggen, H., Pauly, K., Troupin, C., Mineur, F. & De Clerck, O. 2012. Bio-ORACLE: a global environmental dataset for marine species distribution modelling. Glob. Ecol. Biogeogr. 21:272–81.
There are probably some world wide databases but there are also some very good regional long term databases that are public access. An example is the Chesapeake Bay Program which has all that you've listed along with other water quality measures and living resources.
https://www.chesapeakebay.net/
EPA's EMAP program is another resource assuming the data are still readily available to the public given the Trump administrations policies.
Here is the NOAA copepod data base which is a global plankton database:
there is a global data set called Bio-Oracle, which could be useful. It has the aim to be used in marine species distribution modelling, and covers the variables you listed except pH. The dataset contains both present-day and future values until the end of the Century.
http://www.bio-oracle.org/index.php
Scientific references which describe the data:
Assis, J., Tyberghein, L., Bosch, S., Verbruggen, H., Serrão, E.A. & De Clerck, O. 2018. Bio-ORACLE v2.0: Extending marine data layers for bioclimatic modelling. Glob. Ecol. Biogeogr. 27:277–84.
Tyberghein, L., Verbruggen, H., Pauly, K., Troupin, C., Mineur, F. & De Clerck, O. 2012. Bio-ORACLE: a global environmental dataset for marine species distribution modelling. Glob. Ecol. Biogeogr. 21:272–81.
depending on what you are looking for you can also check at the NOAA-websites (http://www.ospo.noaa.gov/Products/ocean/sst/anomaly/) presenting regular global sea surface data and e.g. anomaly charts, providing, among others, forecast data for coral reef bleaching susceptibility in certain areas... this mainly for the temparature records, but updated almost daily.
Dear colleague, the information provided by the other colleagues is very valuable. However, in addition, I would recommend the information available on the following site. Well in addition to the information for the present there are maps available with palaeographic information.