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It is not a good idea to display all data on the map (especially if you have more than 100 000 objects) as point. In openlayers there are heatmap(http://www.patrick-wied.at/static/heatmapjs/) and cluster map (http://openlayers.org/en/v3.6.0/examples/cluster.html). I recommend using one of them In my software I create own "interpolation image" with data
Depends on what you consider "visualisation". You can just use QGIS. Load the coordinates as separated text layer or whatever format you have. Then use the OpenLayers plugin to get a background OSM layer. Or if you actually do have world-wide data, better use Natural Earth vector data to show it in an appropriate projection. http://qgis.org/ ...
I think that with 13,000 records you might still get away with a single app loading a GeoJSON or TopoJSON file. Are you interested in sharing the data? I've got a single page web-app that allows you to explore a bunch of data about crashes here. If you use the clustering plugin, you get better performance. And then you can turn the clusters off when you ...
While qgis2web does not explicitly expose an API, it would be possible to write a python script to load each dataset sequentially, and export it to Leaflet. You would need reasonable knowledge of Python and the QGIS API to achieve this.
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