How is the fuselage of the Boeing 787 produced?
There is a YouTube video showing the method for creating the structure:

YouTube: 787 Composite Construction

This video from British Airways shows the joining process:

Fuselage Section Joining Process which you can see is not welded, but riveted together. The sections are lined up and riveted together using joining plates. 


  Since it is not fabricated as a single piece there are bound to be stresses on the joints. Will it effect/endanger the passengers?


No, not more than any other conventionally constructed fuselage. The joining plates are metal and the structure around them is a metal bulkhead. The 787 fuselage is not a 100% composite construction. 


  Are the same methods used to join the wings?


Yes, if you watch the second linked video in full, you can see the composite wings being bolted to the fuselage structure. The entire wing support structure looks to be a conventional metal structure.


  Last but not least, are there problems due to the joints before and if so, what has been done to rectify these problems?


There are new challenges with any type of new construction technique, here are a few articles that outline the challenges:

NY Times: New Challenges for the Fixers of Boeing’s 787
MIT Technology Review: Boeing's Composite Problem

Boeing has said that the 787 is as easy to repair as traditional aluminum skin aircraft. One case in point is the Ethiopian Airlines aircraft ET-AOP damaged by fire. The procedure to fix it is to cut out the carbon fiber material and put a "cure in place" patch over the hole. In this case other structure had to be replaced or repaired like stringers and bulkheads. This was the first major airframe repair that Boeing had to undertake. ET-AOP took 18 months to repair before being returned to service.