Really not involving studying?
Confucius suggested the Six Arts of education as the scope of what people should study. Therefore, singing can be considered as studying as well (although Music in the Six Arts actually referred to playing instruments).
學 is simply learning
Anyway, the term 學 is the direct translation of "learning". Studying is not necessarily involved. On the other hand, these words have more academic feeling:
Words specifically about learning
- 修 literally means "repair", but has different meanings when paired with other characters (this word carries complicated meaning involving cultural and religious background, and even as a native Chinese speaker I am not confident to independently explain this word)
- There is an ancient saying: "修身, 齊家, 治國, 平天下", which is translated as "complete/improve yourself (your knowledge), lead your family, rule the country, then conquer the world". Refer to this article for details
- 修煉 is used by some religious practitioners or Taoism/Buddhism as a way of improving one's experience and to "accumulate" some kind of concept that is similar to XP in modern games :P (who knows, maybe modern games copied such concept). Note that 煉 has a 火, which means refining of metal, and Chinese people often copmare refining of metal to refining of one's spirit/life experience.
- 修練 simply means practising. 練 means practising itself.
- 習 (revise)
- 讀 read
- 書 means "book", and 讀書 literally means "read books", but also refers to the generic action of going to school in modern Chinese
- 研究 (research, explore, investigate)
Translating the three examples in OP
I learned to sing well in church.
- I, in church, learnt singing songs, which made me sing so well.
- This assumes the condition that you were just singing and you were talking about your performance, as you see from "such a good way".
- This also seems odd in Chinese.
- I, in church, learnt to sing songs well.
- This seems to mean something different from "sing well".
I am unable to make a natural translation that does not change the meaning of your English sentence. Probably because of the culture of humility, in Chinese, we rarely say ourselves sing well in such a way -- children usually get scolded immediately whenever they say that they are good at something, especially without citation or comparison, or without a specific condition like this one.
I learned to walk at age two.
Here, "學會" refers to acquiring a skill.
Where did she learn to manipulate people like that?
她從哪裏學會那樣作弄 (I don't understand the term "manipulate" well enough) 別人?
I would like to refer to a story:
Mencius originally lived near a graveyard. He and his young friends always see people mourning when burying, so they imitated (學着 as in modern Chinese) the people. Mencius's mother is upset with this and moved near a market, but he and his new friends imitate (also 學) how hawkers yell and advertise instead. Eventually they moved near a school, where Mencius and his new friends imitate how students read aloud.
In this classical story (I realized that citation is seriously needed, but I am unable to find any reliable sources even from Wikipedia! This is probably a myth), 99% of the sources describe all the three actions as 學, but the former two are obviously not involved in studying, but the meaning of "learning" as in "imitation".
Studying in Chinese culture
After all, Chinese culture, especially the academic part, is totally about studying. The only difference is whether you are studying books (讀書). There is a saying "萬般皆下品，惟有讀書高": "everything is bad/low-class/low-quality, and only studying books is high-class/good". This proves that Chinese people considering everything as learning, but studying books (especially, studying classical books like Analects) is a different level.
Therefore, in Chinese culture, the main difference between "studying" and "learning" is whether it is the publicly recognized as high-class studying, i.e. 讀書, because for more than 1000 years, Chinese people had to study classical books to enter the exam to become officials, which is considered the common aim for most scholars.
(Yes, keep in mind when looking for translation of words that cultural difference also matters)