I'm from Hong Kong, so I base on HK Cantonese to answer you.
1.教員、教師、老師、先生 all mean teachers. 教員 means members of the academic staff e.g. 教員室(teachers' room), 教員數目(number of teachers). 教師 refers to the profession or occupation. 老師 is the person who teaches you. So my teacher is 我嘅老師. 老師 is also used to address a teacher, e.g. 陳老師、李老師. 先生 is used the same way as 老師 but is old-fashioned. As 先生 also means Mr, e.g. 陳先生(Mr Chan), it would be embarrassing to call Miss Chan 陳先生. The most common way to call the teacher Miss Chan is "Miss Chan".
2. 你教邊科㗎？(Which subject do you teach?) 你教乜嘢㗎？(What do you teach?) . Ending your question with a soft "㗎" would make the listener more willing to answer you. "你教乜嘢書？" sounds like "What the hell are you teaching?" though I know you don't mean that.
3. Both 細文仔 and 子女 can refer to one's own children. However, 細文仔 is very old-fashioned. 子女 is a formal term, which is more used in written form. The most common term for young child/children in oral conversation is 小朋友, literally "little friend". e.g. 你有無小朋友呀？(Do you have kids?), 幼稚園有好多小朋友(There are many kids in the kindergarten). Older kids are called 細路、細路仔(boy) or 細路女 (girl).
There are many more, like 男仔、女仔、兒女、兒童……take your time to learn them.:)