Convert only if you need compatibility with very old software or specific legacy applications that don't support PNG. PNG is superior to BMP in almost every way - smaller files with the same quality. Most modern software prefers PNG over BMP.
No quality loss - both are lossless formats. However, BMP files will be 2-4 times larger than PNG for the same image. You're trading disk space for legacy compatibility with no quality benefit.
BMP files are typically 200-400% larger than PNG. A 1MB PNG might become a 3-4MB BMP. PNG's compression algorithm is very efficient, while BMP stores pixel data completely uncompressed.
Standard BMP doesn't support transparency, while PNG does. If your PNG has transparent areas, they'll be replaced with a solid color (usually white or black) in BMP. This is another reason PNG is superior.
No need. PNG is lossless like BMP but much smaller. All modern image editing software supports PNG excellently. Use PNG for editing workflows - you'll save disk space with identical quality. Use BMP only if specific legacy software requires it.