Trim the fur out of the seam allowance before stitching, whatever machine you choose. You shouldn't need to finish the seams, so I'd use a narrow zz to stitch it, just enough to allow a bit of stretch in the seams. I've had good results with a regular straight stitch and a walking foot, too. But, if you trim the fur really short, you can probably run it through your serger.

If the backing is rough, you won't want the seams to rub against her skin either, so I'd say separate lining. It would it be easier to just put her in leggings and a long-sleeved turtleneck under the costume--that's what we used to do in the theater when I worked there.

Annie

If it is long hair fur, the best way to cut out the pattern is to draw the pattern on the back of the fur, then with short blade scissors, cut the fabric from the back, not putting your scissors far into the fabric so that you are cutting off gobs of fur.
Oh yeah, it will be soooooooo messy, fur everywhere. You'll be eating fur!
Kathy G in MI