Satin can be evil. I avoid it when I can. Be certain to prewash all your fabrics to ensure that your cosplay is machine washable - dry cleaning is too expensive! If it's been washing in washing machine prior to turning into an outfit - it'll be washable after.
For the fraying edges: do you have a serger? This is a PERFECT excuse to purchase one. Sergers will finish the edges of any fabric and keep it from shredding. The other options for controlling fraying for satin would be to do a Hong Kong finish on the edges (this translates to attaching bias tape to the edges of the fabric) or to do French Seams. (Note: you can look up "Hong Kong Finish" and "French Seam" for instructions on how to do these - I think Sandra Betzina has instructions on her site...). At the very least, get a pair of pinking shears and trim the edges with them - it's better than letting it shred to death. Also, be sure to line the outfit if you're just pinking the edges (but still be prepared to discover that it's coming apart - Satin does that unless it's controlled by a serged edge)
To keep it from slipping, there are a few possible solutions
- use temporary adhesive to glue together the two pieces before stitching with machine
- BASTE pieces together by hand and then sew by machine. (better choice, but much more work)