How to create and manage marker, Line, Rectangle, Polygon and route.

The osm API does not have that kind of features. Its purpose is to edit the map data, not to draw markers, polygons, etc on the map.

To do the later, see (amonst others) the leaflet or openlayers libraries. Some third-party websites such as umap or mapbbshare also do somthing very similar.

If you're looking for a place to obtain data from prior to displaying it in an on-screen map, the OSM API probably isn't the best place (it's intended to be used for editing, not creating maps from). Instead, perhaps Overpass might be a place to look. It'll get you exactly the same data, but in a way that doesn't overload OSM's servers. You can also of course just download the data.

With regard to routing, you're probably best just starting from the wiki page. There are offline routers available that you can deploy, and existing instances of them. If you want to do routing in-client you'll probably find some pointers there for that too.

found a new one:

have a look at http://geojson.io and read its help text. You can draw geometries, display markers according to coordinate tables, overlay GPX or KML data, and finally share your map even privately.