To start off, the prohibition on drawing living things is contested by respect scholars, most important among whom is Shaikh Yusuf al-Qaradhawi, president of the World Union of Islamic Scholars.
As to your question, while we can't define soul, this question can be assumed to ask: "What things have agency?" (i.e. free will, accountability for their actions). To answer this, the only (or most) reliable source that we have is the Quran. We know that humans and the jinn have accountability.
Besides this, one curious hint toward the accountability of other things in the Quran is Solomon saying to the bird: "We will see whether you said the truth or you were among the liars." (27:27) And before that he says he will give a severe punishment to the bird, or execute it, for not appearing at the gathering he was holding. Thus Solomon considered the bird accountable for its actions.
This however is not proof that animals have accountability. One could say that that particular bird was specially made (unlikely due to the context of verse, which suggests that the bird's accountability is a matter of course, that it is not a special creature), or that the objects under Solomon's will were given souls temporarily for his sake.
This is more supportable due to the Quran mentioning mountains and birds as being given a special nature for the sake of the kingdom of David (and his inheritor, Solomon): "We favored David with excellence, (and commanded): 'O mountains and birds, glorify the greatness of God with him.' And We made iron pliable for him."
So the issue is not decided on animals and other creatures.
As for the angels, their having agency is not a certain issue. Shaikh Ahmad Mustafa al-Maraghi in his tafseer, Tafseer al-Maraghi, mentions the many views on this topic, such as those who say that the angels are completely good and have no ability to disobey God, and others who said that goodness is their nature but they can disobey, the same way that for the jinn and humans the following of desires is their nature, but they can choose to obey God. God made the angels prostrate to Adam, which some have considered to show that a (good) human is better than angel, since, they say, angels have no desires and temptations to fight and goodness is their default, while a good human has fought to be and stay good.
This is all the knowledge that has been given us, I don't think anything more can be found out on the topic. You can dive into hadith to try to find more answers, but you will likely find much contradiction and more questions than answers, since hadith is a human work and subject to all the shortcomings that comes with that, and when it comes to issues of the Unseen, the world is filled with weak and questionable ahadith.