The image below is characteristic of MS-DOS and NES games. What is this kind of pixel art called, and how to paint on a similar canvas where pixels take on a diamond sort of asterisk look (*). 8-bit comes to mind, but it's just a number and meaningless since many of these games were higher bit by the early 1990s
Short answer: This specific example is (low-resolution) EGA pixel art (with the default CGA palette), and that's what I would call the style if I saw it. NES pixel art is identified either as '8-bit pixel art' (see below) or simply as 'NES pixel art'. You can use a search engine to see more examples of each style, and find tutorials to match them.
Less short answer: The recognisability and 'feel' of both of the styles heavily depends on the exact palette. This is both the amount of different colours available, and exactly what colours are available. For both NES and early EGA, the palettes were fixed, and they are part of the style's identity. Early EGA was limited to 16 standard colours, confusingly called the CGA colours. NES had a palette of 54 colours, but could only display 25 of them at once without clever coding tricks. Wikipedia has an extensive article about palettes and other graphical limitations for just about any hardware.
Another possible factor to successfully emulate a style is post-production aimed at emulating the older screens these styles were originally meant for. Blurring effects and the addition of artificial scanlines are only the most important examples.
Bitcount, like 8-bit, is used to indicate what kind of hardware a piece of pixel art is borrowing its style from. As the NES was an 8-bit machine, its pixel art style get the same name. 16-bit would be Super NES.
To confuse matters further, 1-bit and 2-bit usually don't refer to the machine type. Rather, the number represents the amount of bits used to count distinct colours in the palette. 1-bit is black and white, 2-bit is 4 colours (eg. CGA).
The 'asterisk' effect you see here (I'm sorry, I don't) is most probably due to the dithering effect, especially in the greys of the houses. This is a checkerboard pattern of two different colours to create a third; a common trick to circumvent the limitations of the palette. Blurring on then-current CRT screens took off the worst of the checkerboard effect we see now.
While dedicated sprite editors are handy, I'd recommend using Illustrator (or other vector editing software) to draw pixel art. That allows you to retain the look of pixels but gain the benefit of a resolution-independent source.
Adobe has a starter page with tips on drawing pixel art in Photoshop or Illustrator (also applicable to free alternatives like GIMP and Inkscape), but generally use a square grid, enable align-to-pixel mode, and use Live Paint/square-brush tools.
Most people will just call this 'Retro', 'lo bit', or '8 bit' but Vincent's answer provides good detail about the specifics of the palette.
There is no such thing as asterisk shaped pixels -- a pixel is the lowest resolution element in a display, such that it can only be rendered with a single color. These elements are arranged by a square grid in all popular displays, so the pixels are also square.
The asterisk effect that you're noting is actually a technique called dithering. Artists use it in a limited palette to create a new half-tone without needing to add a new color to the palette. It's a digital analogue to Ben Day dots and half-tone printing techniques for newspapers and comics.
For pixelling software, you can reasonably use anything. In my experience Asesprite's toolset is robust and pared down to pixel art needs compared to anything else. GIMP is a nice free tool after minimal tweaking, and you can write up Python extensions if you so choose. I take a hard stance against Photoshop and Illustrator as pixel art tools, because they're expensive and their expensive features don't support pixel art. There's discussion above about vectorizing pixel art using illustrator and photoshop, but that's something you can do much better by finding algorithms online dedicated to pixel art (and even better than that if you take the programming plunge and code it to your needs)