Please make sure that bikeways and streets are actually connected at the connection points.
In addition, you will need to introduce DriveTime, BikeTime and PedestrianTime attributes in the network dataset. Depending on what type of time you want to use, set the corresponding attribute as the Impedance in the NA layer properties.
Please find more details here:
But if I put a start on a bike route and the end on the streets, it says there is no path.
I had a similar issue a recently, but it wasn't a result of a Z elevation issue. I had created a network dataset using snapping, but I still wound up having network connectivity issues. I primarily use ArcGIS Pro, but a good tool to use to inspect network connectivity is the Network Identify tool in ArcMap.
Additionally, there's a helpful blog and downloadable resource of ArcGIS Pro tasks to create a local government network dataset which might be of use in addition to the Paris multi-modal network tutorial.