If you notice, the blocks all have a solid line around the edge, then a dotted line. This is to help you square up the block. The dotted line is 1/4" seam allowance. Do not remove the paper until the blocks are joined together. As you said, some of the pieces in the blocks are bias cut. By not removing the paper until everything is joined together, you avoid stretching and making the blocks "wonky". Yes, it is a lot of work to remove all the papers at the same time, but I always do this in front of the TV. I've made queen sized paper pieced quilts and removed the papers after the entire top was put together. Took awhile, but kept the blocks square.
The method I use is probably similar to the fold and cut method mentioned above. I sew on the line indicated on the block using a generous amount of fabric. (I never precut fabrics, I just make sure the piece covers the # on the block I want to sew). I then fold the paper back onto the sewing line (paper is on top) I just sewed. Using a ruler, put the 1/4" mark on the sewn line (where the paper is folded back) and rotary cut off the fabric that hangs over the ruler edge. (I hope this is clear). Press the fabric with the paper still folded back. Now take the paper and top layer of fabric and open them. Press flat. (Clear as mud yet?) Continue on with the next piece in the block. Hope this helps!