Building map with custom roads & buildings for a forest?

There seem to be both technical and legal aspects to your question. In principle, adding paths or tracks to OpenStreetMap isn't an issue provided they actually exist on the ground and the source for your information is compatible with the OpenStreetMap license. I mention the license here because you say you have a shape file for the data, but don't mention the source. I know very little about the legal side of things, but I think the copyright holder would have to give express permission for the data to be included in OSM and published under OpenStreetMap's licensing terms. There are Import Guidelines for this.

Adding streets to OpenStreetMap is fairly simple and many editors are available, the most popular being the in browser editors iD and Potlatch 2 (available on the main site) and JOSM (standalone download). Some of these have shapefile support if you are sure you have proper permission to use it. As you put the word "streets" in quotation marks, I think it might help if you look at the highway page on the wiki as they might be better classified as a track or path in OSM terms.

Your "feeding boxes" sound like they might be either "vending=animal_feed", "amenity=feeding_place" or "amenity=game_feeding".

There are several starting points for creating apps depending on whether you are creating something for Android, iOS or the web. Some keep all the map data on the device (probably sensible for something the size of a forest) and so may not need a tile-server or any internet connectivity at all while they are being used. Others rely much more heavily on a constant internet connection. There are also commercial providers of map tiles and back end services based on OSM that might work out cheaper than running your own server. I'm afraid I can't give any specific recommendations as I'm not a developer myself and am still a little unclear on your use case.