If you can find some, the same wood is always best as it should expand and contract at the same rate as the body so will add minimal stress in the future. But it’s not a major problem, so some generic hardwood dowel should be fine.
Yes, the binding takes up stain (especially spirit-based stain). The surface can be scraped and cleaned, but you’ll find lots of hairline cracks on the small radius bends, and the stain here remains. It may be the binding material the kits use as Gibson etc. stain and scrape without problems.
Various methods have been used to stop this issue, and covering the binding with clear coat first (masking off the body) is probably the most effective. Masking the binding helps, but it is still prone to seep under the binding at the edges. Clear coat and then mask would be my advice. You may need to scrape or sand off the clear coat if it itself stains, but it should protect the binding underneath.