I’m new myself. It sounds like you are using the online editor (Potlatch), which seems to be only for basic editing. You can add a toilet by adding a different amenity like Parking then replacing the word parking in the value box with the word toilets, it should be in the popup box that appears. I would recommend reading the wiki and trying one of the offline editors.
If you are using potlatch, click twice on the map to create a green dot then click the little plus sign at the bottom right, at the bottom left you will see a new entry with “key” highlighted. Replace the word “key” with “amenity” then in the box to the right of that, type “toilets”.
No, most POIs/nodes in Potlatch are represented as a green dot. The only POIs which have thier own icon are the presets you can see at the bottom of Potlatch.
The POI on the map will be rendered with an icon, if the renderer supports this. There are many type of POIs which won’t be rendered in Mapnik (the default OSM-Map). Osmarender (try the blue “+”-Sign on the OSM-Homepage) has other preferences of which type of POI should be rendered. If you search the internet you will find many other sites providing maps based on OSM-Data. Depending on thier preferences you will find rendered icons of POIs you won’t find on other maps.