The beading usually would not catch the edge of the applique so I would generally not suggest leaving the edge raw & effectively straight stitching the applique down (via embroidering on the beads) unless you want your applique to fray quite a bit more in that one area.
As for the rest of your questions, either a photo or a more detailed description of your process & the outcome are necessary for us to provide any useful advice. I can try to guess based on what you've written, but I've made some assumptions that might be wrong.
(1) It sounds like you appliqued both the chaise & the cat using raw-edge machine applique. You mentioned "stabilizer". Stabilizer is for embroidery or for adding structure. For a quilt, you want to use Fusible Web (WonderUnder, Heat & Bond). I doubt the "puckers" will come out if you are knotting through a stabilizer, but I don't know. Depending on the stabilizer & how big the puckers are, perhaps doing the Trapunto that Tartan suggested will help to pull them out. Of course, Tartan is a really talented & experienced quilter. The rest of us rely on wash-away thread for our Trapunto because otherwise it is really obvious we went back over the same line twice & didn't come close to hitting the stitches in the same place.
If you actually did use Fusible Web, let me know & I can tell you what I've done to get the bubbles between the layers of fabric out.
2. Whenever possible, when I'm layering applique I create applique units. I take the top layer & applique that to the layer of applique below that & then attach the whole "unit" to my background. It means I don't have to stitch through so many layers of fabric. I do this whether hand or machine appliqueing. Assuming you are attaching the beads by machine, I would applique the cat to the chaise, embroider by beads on, (add the chaise legs) & then attach the entire unit to the background & add trapunto. Once you've finished the trapunto, sew tufting through all the layers.
3. I'm not sure why the original chaise didn't have legs attached. If you end up remaking it, I'd draw some legs on. So much easier. It will not look right to have the rest of the chaise sewn raw-edge & then do a turned edge finish on the legs. You can fuse the legs to the background fabric & hand stitch them down if you don't think you can stitch them on your machine ... or you could embroider them -- just make sure to do that before adding anything puffy behind the rest of the chaise as I would think the added bulk might impact how nicely the embroidery turns out.
Please do post a picture so we can help better. It sounds like a really adorable design!