Not sure what specific errors you are referring to but there is no tag to add to the waterway if it crosses a bridge as it is implied that bridges cross above waterways. Any culverts I add, I always add the layer tag (-1) to the waterway.
As I am adding waterways from a dataset and satellite imagery, it is not possible to add how the waterway crosses the roadways, especially culverts, tunnels and fords and even bridges are not very distinct in country areas. This needs to be done later by other mappers doing surveys. Where others have mapped culverts or ford or combined culvert and ford at some crossings, I have preserved those.
Due to the large amount of data, I would be guilty of not zooming in close to every crossing to ensure the dataset aligns with satellite imagery closely but from doing so much of this I notice I am not alone. 
If waterways are mapped crossing road or track infrastructure without bridges, culverts, tunnels or fords it is not an error, it is an indication that no one has surveyed the area yet. If Osmose is picking these up, I would expect them to be considered an alert and not an error.
No doubt that I have made a few errors whilst entering so many waterways but am using the Josm validator before upload and also using Osmi and the WaterwayMap later.
I'll have a look at Osmose too when I finish the catchment.