Does Islam believe in souls (spirits)?
Yes, it does believe, and it's mentioned in many places, I would mention this:
وَيَسْأَلُونَكَ عَنِ الرُّوحِ ۖ قُلِ الرُّوحُ مِنْ أَمْرِ رَبِّي وَمَا أُوتِيتُم مِّنَ الْعِلْمِ إِلَّا قَلِيلًا
Muhsin Khan Translation
And they ask you (O Muhammad SAW) concerning the Ruh (the Spirit); Say: "The Ruh (the Spirit): it is one of the things, the knowledge of which is only with my Lord. And of knowledge, you (mankind) have been given only a little."
Does Islam believe that Humans have souls?
Yes it does, and It's clearly mentioned in Quran that Allah gave Adam a spirit when he created him as comes in this Ayah (in which Allah is talking):
فَإِذَا سَوَّيْتُهُ وَنَفَخْتُ فِيهِ مِن رُّوحِي فَقَعُوا لَهُ سَاجِدِينَ
Muhsin Khan Translation
"So, when I have fashioned him completely and breathed into him (Adam) the soul which I created for him, then fall (you) down prostrating yourselves unto him."
Does Islam say that women DON'T have souls?
NO, there is nothing in real Islam that says that women don't have souls. Islam respects women very much and orders men to treat them very nicely.
Does Islam say that women DO have souls?
Yes, again and again I'm surprised how Qur'an is really miraculous and eloquent! Qur'an contains an ayah that internally provide the fact that women have souls! in fact I never ever thought of that before when reading this ayah because I never imagined that one may request authentication for whether women have souls or not (and the more amazing thing is that it's the fist aya in Surat An-Nisa', which is Arabic for: The Chapter of Women).
يَا أَيُّهَا النَّاسُ اتَّقُوا رَبَّكُمُ الَّذِي خَلَقَكُم مِّن نَّفْسٍ وَاحِدَةٍ وَخَلَقَ مِنْهَا زَوْجَهَا وَبَثَّ مِنْهُمَا رِجَالًا كَثِيرًا وَنِسَاءً ۚ وَاتَّقُوا اللَّهَ الَّذِي تَسَاءَلُونَ بِهِ وَالْأَرْحَامَ ۚ إِنَّ اللَّهَ كَانَ عَلَيْكُمْ رَقِيبًا
Muhsin Khan Translation
O mankind! Be dutiful to your Lord, Who created you from a single person (Adam), and from him (Adam) He created his wife [Hawwa (Eve)], and from them both He created many men and women and fear Allah through Whom you demand your mutual (rights), and (do not cut the relations of) the wombs (kinship). Surely, Allah is Ever an All-Watcher over you.
Although the translation translates "نفس" as "person", the word "نفس" in Arabic is mainly a synonym of "soul", as comes at the very beginning of the explanation of the word "نفس" in the Major Arabic Dictionary Lisan Al-Arab (The Tongue of Arabs):
النفس : الروح
النفس (the نفس): The soul (spirit).
Lisan Al-Arab - The Letter Noon - The section of نفس
And there are many other quotation that prove they are identical, I won't quote because the answer would become long, some can be found either in the same link of Lisan Al-Arab provided above, or from this link.
So that ayah clearly says that the "wife" of the first soul was created out of it, and then from both many men and women where created.
Furthermore, the hadith that talks about the creation of babies never specifies that he is a male, it's general:
'Abdullah bin Mas'ud (May Allah be pleased with him) reported:
Messenger of Allah (), the truthful and the receiver of the truth informed us, saying, "The creation of you (humans) is gathered in the form of semen in the womb of your mother for forty days, then it becomes a clinging thing in similar (period), then it becomes a lump of flesh like that, then Allah sends an angel who breathes the soul into it; and (the angel) is commanded to record four things about it: Its provision, its term of life (in this world), its conduct; and whether it will be happy or miserable. By the One besides Whom there is no true god! Verily, one of you would perform the actions of the dwellers of Jannah until there is only one cubit between him and it (Jannah), when what is foreordained would come to pass and he would perform the actions of the inmates of Hell until he enter it. And one of you would perform the actions of the inmates of Hell, until there is only one cubit between him and Hell. Then he would perform the acts of the dwellers of Jannah until he would enter it."
[Al- Bukhari and Muslim].
وعن ابن مسعود، رضي الله عنه ، قال: حدثنا رسول الله صلى الله عليه وسلم، وهو الصادق المصدوق: "إن أحدكم يجمع خلقه في بطن أمه أربعين يوماً نطفةً، ثم يكون علقة مثل ذلك، ثم يكون مضغةً مثل ذلك، ثم يرسل الملك، فينفخ فيه الروح، ويؤمر بأربع كلمات: يكتب رزقه، وأجله، وعمله، وشقى أم سعيد. فوالذي لا إله غيره إن أحدكم ليعمل بعمل أهل الجنة حتى ما يكون بينه وبينها إلى ذراع، فيسبق عليه الكتاب ، فيعمل بعمل أهل النار فيدخلها، وإن أحدكم ليعمل بعلم أهل النار حتى ما يكون بينه وبينها إلا ذراع، فيسبق عليه الكتاب فيعمل بعمل أهل الجنة فيدخلها" ((متفق عليه)) .
Where does this notion regardless of its truthfulness come from?
Not sure about it, but this might have come from the beliefs of Alawites (who are an extremely distorted sect of Shi'a and are not considered Muslims by much of scholars), I read once that they believe in reincarnation: If a man (not woman) had a good religious life, his spirit will reincarnation as another man, if he had a bad religious life however, his spirit will be damned and cast into a woman, and then into an animal, this (if was true) may be the primary source of that notion about women not having (proper) souls.