- Serving map tiles
Tilestream is nice but if you want to have live maps (showing new OSM data as they happen) you need Apache and mod_tile (I think this is also used in main OSM site). You download OSM extract from geofabrik or anywhere else set up postgresql server. And updating mechanism. And them every time tile is requested it is checked if data was changed in DB and if it was tile is re rendered otherwise it is just returned. DB is updated each minute/hour/day how fast do you want it. You have to use mapnik style with this but Tilemill can export mapnik xml style. A lot of Tilemill styles are also on GitHub.
- Updating OSM Data
iD editor is main OSM editor for beginners and all changes are saved to OSM database as soon as you save them. It can take couple of minutes to see them in map tiles. This is because of caching. But if you use minute planet diffs you get changes as soon as you made them.
The same is true for all other editors (Potlatch, JOSM, Merkaartor). JOSM is used for more advanced editing usually. It has many more tools. Next to Bing there is also "Mapbox Satellite" option for tracing in iD and JOSM. This are images directly from DigitalGlobe. But it is always better to use data from GPS than only tracing images.
For mapping coordination you can use OSM quality assurance
JOSM has many plugins used to speed up drawing. One very useful for Public transport mapping is Public Transport You can drive with a bus and set waypoints each time you are on a stop. Then at home plugins helps you to add stops to OSM. It also eases adding routes. After all of this is in OSM you can see it in Transport style.
For mapping transit GO Sync should be useful. I also tried OSM2GTFS for bootstrapping GTFS but it didn't work for me on Linux. I write something on my own but didn't open source it yet.
- Routing engines
Why would you need 2 routing engines? I'm currently using OTP for my city routing and it's quite nice. It has multi-modal transport routing, pedestrian routing, bicycle routing and car routing. It also uses elevation data in routing if you have them. (You can get them from STRM or similar) OTP is a router for shortes/fastest path and transit. For example:
OTP is a java REST server with Leaflet fronted.
There also exist opensource Android application to access OTP server. It is also on Google Play and you can add your server URL to connect to it.
I think only OTP and Graphserver support OSM and GTFS data and are open source so you can add own data all others doesn't support Transit data.
You are not using geodesic functions to calculate the length, which means that for a point there is an error factor of:
cos( LATITUDE * pi() / 180 )
If you then multiply the calculated lenght by the error factor you should obtain a value pretty close to the actual trail length. For instance the Old Beechy Rail Trail is close to Melbourne, which has a latitude of ~ 37° which means:
cos( 37 * pi() / 180 ) ~ 0.798635510047293
in turn:
60 * 0.798635510047293 = 47.91 Km
which, depending on your needs and given the brutal rounding of Melbourne's lat that I took in the calculations, could be considered close enough to the reported official length of 46 Km.
This question has very useful information on the topic of geodesic measurements.
Best Answer
There are two broad ways to do this - one is to piggyback on someone else's infrastructure. For instance, the overpass turbo api has a query already baked in to get cycle routes and this can be exported as geojson (see "load" tab). This is great for one-off queries and where you have a small amount of data, but you are reliant on the generosity of others so you need to bear this in mind morally and from a practical point of view (there is nothing stopping them turning off the server after all). The overpass API is just one of many similar services, including pre-built shapefiles etc, but in my opinion it is a good one to use.
The alternative is to create your own infrastructure. This is not a small undertaking, and basically involves replicating the main OSM database (or subset of) and creating your own images. There are a large number of guides (e.g. here) to do this and I can't possibly do the subject justice here, suffice to say that it is generally quite hard and will require a reasonable level of technical expertise. You also need to work out the difference between map tiles and vector data (geojson) and which you want to serve.
You could also "buy" in the tiles from a 3rd party supplier (e.g. Mapbox) and then just keep a database of cycle routes to serve as vector data. That would probably be simpler, but still involves syncing databases etc. Something like Osmosis could be used but there are many alternatives.
There are literally hundreds of different ways this question could be answered so I suggest you are a bit more specific about what exactly you want to do.