She has studied some knowledge; can she teach it to people?
Question

I have not studied shar‘i knowledge in full, but I have studied some basic issues of Tawheed, Tawheed al-Ruboobiyyah and Tawheed al-Uloohiyyah, the Divine names and attributes, and other issues mentioned in the book Ma‘aarij al-Qubool by al-Haafiz al-Hakami. Then I learned from the shaykh here some introductory points of hadeeth science and some topics of fiqh, then some study of al-Nawawi’s Forty Hadeeth and some raqaa’iq (heart softening reports). Then the shaykh left us and I took on the mission of teaching shar‘i knowledge in the mosque here to the women. Is there any sin on me? What should I do in order to become qualified for this?.
Praise be to Allah.May Allah reward you with good for your interest in learning about your religion and teaching people. If your intention is sincere, we give you the glad tidings of a great reward with Allah. The Prophet (blessings and peace of Allah be upon him) said: “Allah, the angels, the inhabitants of heaven and earth, even the ant in its hole and even the fish in the sea, send blessings upon the one who teaches the people good.”
Narrated by al-Tirmidhi, 2609; classes as saheeh by al-Albaani in Saheeh al-Jaami‘, 1838. 
By the grace of Allah, the one who teaches the people something about their religion deserves this supplication, because what he taught them will bring them to something good. 
In Saheeh al-Bukhaari (3461) it is narrated from ‘Abd-Allah ibn ‘Amr (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “Convey from me, even if it is a verse.” 
Al-Haafiz Ibn Hajar said in al-Fath: And he said in the hadeeth: “even if it is a verse,” meaning just one, so that everyone who heard it would hasten to convey whatever verses he knew, even if it was a little, so that by this means everything he (the Prophet (blessings and peace of Allah be upon him)) brought would be transmitted. End quote. 
But the one who takes on the mission of calling people to Islam has to have knowledge and insight into what he is calling them to. It is not essential for him to know the entire religion. Explaining this issue, al-‘Allaamah Ibn ‘Uthaymeen (may Allah have mercy on him) said: 
If a person has proper understanding what he is calling people to, it makes no difference whether he is a great scholar or a seeker of knowledge who is a hard worker, or an ordinary Muslim whose knowledge of a particular issue is clear and certain. The Messenger (blessings and peace of Allah be upon him) said: “Convey from me, even if it is a verse.” He did not stipulate that the daa‘iyah (one who calls people to Allah) should have vast knowledge; rather he stipulated that he should have knowledge of what he is calling people to. As for the one who is ignorant and calls people on the basis of emotion, that is not permissible. 
Hence we see some brothers who call people to Allah although they have only a little knowledge, and we find that because of the strength of their emotions they declare to be haraam things that Allah has not declared haraam, and they impose obligations that Allah has not enjoined upon His slaves. This is a very serious matter. Allah says (interpretation of the meaning):
“And say not concerning that which your tongues put forth falsely: ‘This is lawful and this is forbidden,’ so as to invent lies against Allaah. Verily, those who invent lies against Allaah will never prosper.
117. A passing brief enjoyment (will be theirs), but they will have a painful torment” [al-Nahl 16:116, 117].
End quote. Fataawa ‘Ulama’ al-Balad al-Haraam, p. 329 
Based on that, we encourage you to continue what you have started of seeking knowledge and conveying it to people. Whatever Allah makes it easy for you to learn of matters of ‘aqeedah (belief) and ‘ibaadah (worship), convey it to people as you have learned it. Seek help in continuing your pursuit of knowledge by means of recordings of trustworthy scholars, of which there are many available on some websites, praise be to Allah. Try to listen to the recordings of Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him), for there is much good and great knowledge in them. Similarly, seek out whatever you can listen to or read of the words of scholars who are known for their knowledge and sound belief. 
May Allah help us and you to do that which He loves and which pleases Him. 
And Allah knows best.