All these can be differentiated by having deeper understanding of Kanji.
But to simplify it:
監督 - Movie directors (映画監督) comes in mind in most of the time, usually means a single person that is assigned a task to control. Or 監督する to take control, usually in an environment/situation.
管理 - Has a very wide meaning of to control/management. Example 監督 is ...
表立つ is almost always used in the adjective form 表立った to modify a noun that follows it, or in its adverbial form 表立って. It means something has an outward sign, and it is often used in a negative sentence.
There has been no visible movement so far.
表沙汰 is usually used in the form 表沙汰にする (not * 表沙汰する) or 表沙汰になる to mean, respectively, someone ...
Perhaps even without those options, many native speakers can instantly give 人懐っこい or 人見知り(を)しない as the most natural expression which fits in that blank. 人懐っこい is such a common adjective to describe a friendly baby who smiles instead of crying when held by a stranger. 人懐っこい is also commonly used to describe a friendly animal. This is a rather simple ...
I would say 受け継ぐ is the most general in meaning and most commonly used. It can be used for a wide range of things, including 伝統 (traditions), 特徴 (physical or character traits), and 財産 (properties).
引き継ぐ is often used for 仕事, tasks or responsibilities at a job that someone leaves behind when, for example, leaving an organization. You take over those things ...
As long as you're learning something from an entity (doesn't necessarily have to be a human being), it can be called "先生" (at least in spoken Japanese).
For example: It is very common for people to say
"Google 先生に教えてもらった" (Google sensei ni oshiete moratta)
when they learn something new via Google.
As long as you are interacting with the said person in its role of teacher, you should use 先生.
Meaning that you won't if the person is your friend, family member, partner or any other relationship that doesn't involve its teacher job, even though it can still be used in this case as a joke, irony or to emphasis the teacher position.
Sorry in advance for my unnatural English. If you don't understand what I'm saying, please point out my mistakes.
In addition this answer is just a guess. This is just my opinion.
As a native Japanese, I feel "人生の中の汚点を教えてください"(without （笑）) would be a very very rude question because it sounds like the questioner assumes the questionee has 汚点 in ...