Underground sections of Streams

You should describe the feature as best you can, using existing, well-used tags where possible. If this results in an odd rendering, this is a problem with that render, not your mapping. Indeed, if you omit detail like this because of an odd rendering result, the rendering will never get fixed because its developers won't have an example to work from.

So yes, map the stream in its culvert wherever possible.

I d agree with Jonathan, map them if possible. A watertransport system is easy, a kind of straight line. But how do you know where a undergroundstream is goiing over a certain distance ? Has it properly been pointed to a series of locations ?

Yes

it is always interesting to represent a stream or any waterway [or any way] from its source to its final destination

but I mean that as a whole way rather than add just another relation because of segmentation by a few culverts or bridges.
So, why not draw culverts, bridges, or any interruption, as additional ways? Ways that use the same nodes as the mainstream to indicate the relation, of course.
And well, this does represents reality. A culvert is not the stream, really. It's something around it. Same for a bridge, sort of, it's concrete supporting the same tarmac as the main road ;-)
Now, even if rendering has to be fixed because of what I'm doing, I prefer coping with it right now ;-)
Streams are usually wholly at layer -1 because they're not tsunamis.
Let's put them at level -2 and draw culverts at level -1. The culverts are going to hide the stream. But shouldn't bridges, be at level +1 as a rendering issue instead of a reality -1?

How does that sound to you?