I extended a river. How can I merge the old part and the new part of the river to one river in ID?
Extending a river would not generate a new part. To extend in iD, click on the node at the free end and use the <-+ tool.
However, it would be normal for a river to be in many parts, so there is no need to join the parts, as long as they share common nodes on the boundaries between parts.
The real problem is that http://www.openstreetmap.org/way/402749044 seems to be flowing uphill and doesn’t seem to connect to the sea. To fix the direction you need to use the << tool in iD. I wouldn’t want to advise on the connection with the sea without knowing what happens in the real world (maybe it goes underground?).
If you want to join parts which just butt together (e.g http://www.openstreetmap.org/way/402748556 and http://www.openstreetmap.org/way/482984484)), in iD, use shift-click to select both and then use the + tool. You cannot join two branches, in this way, as way’s cannot represent branched structures.
Great that worked!