What a BMP file actually is
BMP is Microsoft's original raster format and in normal use stores pixels with no compression at all: three bytes per pixel, plus a header. A 12-megapixel photo lands around 36 MB. An optional run-length mode exists but is rarely produced.
BMPs turn up from legacy Windows software, older scanner drivers, industrial and medical equipment, and Paint on an ancient machine. New files in this format are almost always a sign that something in the chain has not been updated in a long time.
What you get from ICO
ICO is a container that holds one or more small square images, with transparency, for use as an icon. It caps out at 256×256 — anything larger has to be scaled down, and the format is not intended for general pictures.
Every browser requests /favicon.ico by default, with no HTML required, and Windows uses ICO for application and shortcut icons. It is a narrow format that is completely reliable inside its niche.
What changes when you convert BMP to ICO
Both formats are lossless, so nothing is thrown away in either direction — this is a container change rather than a quality decision. What differs is how efficiently each one packs the same pixels, and how widely each is supported by the software you need to open it in.
There is no transparency to preserve — BMP files do not carry an alpha channel, so every pixel is opaque to begin with. Converting to a format that supports transparency does not create any; it only means you could add it later in an editor.
ICO cannot exceed 256×256, so anything larger is scaled down to fit while keeping its aspect ratio. Work from a square source if you can — a wide image gets letterboxed into the square, and fine detail disappears entirely at the 16-pixel size a browser tab actually renders.
Size stops being the interesting question here, because the image is being scaled to at most 256×256 regardless. What matters is that a BMP from an old Windows tool is often already small and square, which makes it an unusually good icon source — the pixels are unprocessed and there is no prior compression to fight.