Assalamu alaikum,
From what I know (and insha Allah it's correct), drawing living things by hand is forrbiden. Unless it is for educational or neccesary purposes. But drawing without the usage of your hand is not haram. This is what my Islamic session teacher told me and what Naik Zakir said. But I highly doubt this opinion so it is safe to stay away from it. Acording to the Malki mathhab, 2D drawings are permitted. Regarding cartoons, anime, video games, and other animated things such as Emojis:
They are NOT haram insha Allah. This is according to Islamqa which is a little bit strict when it comes to things like this.
"The basic principle with regard to adults (and children) watching cartoons for fun is that it is permissible, so long as they are free of anything that is contrary to Islamic teaching" (fatwa 225149)
" Emojis do not come under the same ruling as images that it is forbidden to make, draw, or use, for two reasons:
1 – It contains none of the features of a real face, such as eyes, mouth and nose, and it has no head or ears.
The Prophet (peace and blessings of Allaah be upon him) said: “The image is the head; if the head is cut off, there is no image.” Narrated by al-Ismaa’eeli in his Mu’jam from the hadeeth of Ibn ‘Abbaas and classed as saheeh by al-Albaani in al-Silsilah al-Saheehah no. 1921 and in Saheeh al-Jaami’ no. 3864.
2 – The majority of fuqaha’ are of the view that if something is cut off from an image without which it could no longer live, then it is not a haraam image. For a detailed discussion of this issue and the views of other madhhabs, see Ahkaam al-Tasweer fi’l-Fiqh al-Islami, pp. 224-240.
But we should point out two things:
This usage in chatting on the internet does not come under the drawing of images, rather it comes under the use of images." (Fatwa 110504)
So we can see from all of this is that using images does not come under the making or drawing of these images, video games and animated things are permissible, and they donly not come under the heading of imitating Allah's creation. Regarding the hadiths that you mentioned, drawing is permissible for children and their mothers. I hope that I answered your questions properly. I would also like to mention that cameras are not haram and this is the most correct view. An example of a scholar who allows it is Sheikh Yusuf Al-Qardawi from Egypt. Many of the scholars who forbid taking pictures now permit it and they have Facebook and social media accounts of their own.
