The Chibi big-head style is a direct consequence of low-resolution sprites of the 80s and early-mid 90s. Those sprites were typically 16x24 or 16x32, and the best way for a character to be distinctive, have personality, and display emotion was to devote fully half or more of the sprite to their face and head. At these sizes, you couldn't even afford to outline your sprites, but later examples of this style doubled the resolution to 32x48 or 32x64, and could them afford outlines.

My personal opinion is that the outlines start to break away from the original Chibi style, and move more towards traditional cartoon-like design and animation. Double the resolution yet again, and it becomes very inviting to break away from the style even further. I think you could do good, high-res Chibi-style sprites, but you'd really need to hold strongly to the underlying principles, otherwise it turns out to be something very different, and possibly odd-looking in an uncanny-valley sort of way.

A fellow by the name of Tsugumo wrote an excellent series of articles/tutorials called So you want to be a pixel artist? They've been around forever and have aged very well. I first started reading them around 12 years ago, and they are every bit as relevant today. They're required reading, so go do that now. Wikipedia also has a decent article on spriting.

Many people are able to do excellent sprite work in Photoshop or even in paint, but I prefer tools more tailored to the task. My choice for a long time has been ProMotion by cosmigo (about $80), but many people swear by GraphicsGale (about $20). There's also Pixen (MacOS, about $10), ASESprite and Grafx2, which are free, but the latter two look a little rough around the edges and I don't have experience with either, so I can't comment on whether their functionality redeems them.