OSM is a dataset. That's free and open. There are different tile servers (https://wiki.openstreetmap.org/wiki/Tile_servers) but they each have a different policy (https://operations.osmfoundation.org/policies/tiles/)

In a nutshell: you can't just use them in production for your app. Development shouldn't be a problem.

What you're talking about is precisely the commercial service that Mapbox offers. OSM (+ some extra stuff).

Alternatively, you're free to spin up your own tile server. That way you don't pay anyone else (but you pay your server).

Not necessarily, but it might be faster.

Basically you need map data (in form of tiles) and you need a way to show and interact with your map (e. g. Leaflet).

Here is a summary and one way to do it: https://switch2osm.org/the-basics/

Tiles can be images or vector based. I also recommend having a look at https://openmaptiles.org/ if you want vector based tiles. You can generate your own tileset with this (output is a mbtiles SQLite file based on a open source schema; the schema describes which information go into the database and later can be shown) or download a set of tiles (because e. g. for the world it will take a lot of ressources and a long timeto generate a tileset). There are also servers (like tileservergl or tileservergl-light) that can serve the tilset - not only as vector tiles but also as graphical image tiles.

Raw OSM data can be downloaded for free from https://download.geofabrik.de/

The data itself is free. Current map tiles on tile.osm.org is hosted on donated hardware, and run by volunteer sysadmins. That's not free. they reserve the right to block you for any reason at zero notice. Don't run your business or app relying on that. :)

There are plenty of companies who take OSM data and make map tiles. Those charge money or give it away for free.

No, but you can consider donating and if your app is popular, you should probably switch to a commercial tileserver such as mapbox.

https://operations.osmfoundation.org/policies/tiles/