No cost for using the data. OSM’s servers (i.e. any resources you access from openstreetmap.org) are however run entirely on donations and volunteer contributions. You should not use them for commercial services, but should instead either download the data and run your own tileserver/geocoder/whatever, or use a third-party that does that.
No. But see the ODbL stipulations on “derivative database” for any data that you might add.
All these things have been answered a gazillion times before - please do look at https://help.openstreetmap.org/ and your favourite search engine, which will tell you everything you want many times over.
Good to hear that you’re interested in using OpenStreetMap data.
As Richard has already explained, using OSM data is free under the terms of the Open Database License (ODbL). While this license doesn’t affect your intellectual property rights on software, it can have an impact on rights of data that you mix with, or derive from, OSM data, depending on the details.
You should have a look at the ODbL and the other resources linked in this thread for details. But one major requirement of the ODbL that hasn’t been mentioned so far is attribution: You must make your users aware that you’re using OpenStreetMap data. Have a look at the OSM copyright page for some relevant info: https://www.openstreetmap.org/copyright