On edit six:
4*6*(300^2)*20 = 43200000 bits = 5400 KB = 5.4 MB.
Okay, so, you're right in the multiplication here (give or take a quibble about binary vs. decimal KB and MB). That is the uncompressed byte size of the data which makes up the image.
However, it's not necessarily the file size of the image, because most popular image file formats feature compression (either lossy, like JPEG, or lossless like PNG or many TIFF and RAW-type files), and because the image file will also carry some amount of metadata.
(Also, the bit depth is more likely to be 24, with three 8-bit channels.)
But, what are you trying to do with this result? You mentioned network transfer in an earlier edit, and amount of ink used before that.
Particularly, with digital files, you normally can just look at the height and width if that's what your concerned about, with no need to covert to inches and DPI and back again.
It's also worth mentioning that this usage is of "dots" is conceptually wrong, because we're really talking about image pixels, so PPI might be more appropriate. (See this other answer for more on that.) But the terms are often conflated, including in photography software.
I still think What does DPI mean? should cover everything you need to know here.