How can I create my own tags using JOSM?

There are many more tags than the editors show. E.g. there is https://wiki.openstreetmap.org/wiki/Proposed_features/ISCED , which might be what you are looking for. In JOSM, you can just add those tags.

In case you want to create your own tags, it's best to do research whether someone else had the same idea, search our wiki (e.g. 'osm tag XXX' in a search engine will give you good results.

If you cannot find the tag you need, you can come up with your own. But it's best to document it on the wiki, so others understand what you mean and they might start using the tag as well. There is also a formal process to get a tag "approved", see https://wiki.openstreetmap.org/wiki/Proposal_process

OSM has no layers. To get new tags added to the standard rendering, you have to create issues on https://github.com/gravitystorm/openstreetmap-carto . Do not have high expectations for rare tags to get rendered in that style.

You might better create your own map, umap might be a good place to start. Especially in combination with Overpass, read e.g. http://www.mappa-mercia.org/2014/09/creating-an-always-up-to-date-map.html

Since this is a lot of information to digest, please do not hesitate to come back and ask more specific questions (preferably as new questions) here on this site.

Good luck

There are quite a few questions here, so my answer is quite long.

To answer your last question first:

OSM has not notion of layers for different types of data (hospitals, schools etc). Everything is held in a common data pool. This may make accessing different types of data a little harder (but see below), but comes with the immense benefit of showing explicit inter-relationships between different objects. Not only can be have a school, but also show which entrances in a school are suitable for a wheelchair user; or enable choice of routes to school which minimise crossing busy roads.

So when adding infrastructure, even if you add schools, hospitals separately, ultimately they should get interconnected as more detail is added.

Now on to what can be done in JOSM:

In JOSM it is perfectly possible to start a separate data layer for each type of object. These remain independent until you load them up to OSM.

Additional tags can be added directly in the tag edit panel once you have created the basic element (e.g., school). JOSM will also remember which tags & values you have used recently. It is also possible to configure JOSM to provide specific presets relevant directly to what you are mapping: e.g., to show school types and age ranges. Indeed, suitable preset configurations may already exist.

For retrieving data relating to a particular group of objects, the best tool to use is Overpass-turbo, for instance here schools.

Finally note that there are already widely used tags for indicating school type (primary, secondary, etc.) and age ranges of the school children.