To "keep something on" and to "leave something on" mean virtually the same thing. Either can refer to an article of clothing that one intends to (or is admonished to) persist in wearing, or to some kind of powered machine (a car, a refrigerator, an electric light, etc.) that one intends to or is requested to permit to remain in the running state.
Both of those can also take "off" instead of "on" as a complement to turn the meaning around into a negative.
Please keep [or leave] the light off. I'm trying to sleep.
To "hang on to" something is a bit different. It can mean one intends to or is requested to not get rid of something.
Hang on to your hopes.
Hang on to that receipt. You'll need that if the IRS audits you.
"Hang on" can also be a request for someone to wait.
Hang on! I'll be there in a minute.
Note that using "off" changes the meaning into something that is not the opposite. It means to literally suspend something or someone from something else.
I wouldn't want to hang off the side of a building until we see world peace.
"Leave off" also can refer to a point where someone stopped doing something:
Where did I leave off in the story I was reading to the kids?
And "keep off" can be a warning against trespassing:
Keep off the grass!
This is a difficult question to answer, because both aches and pains are subjective experiences - like colours - which you're unable to share, but assume everybody understands. I would have assumed that every language has words for both ache and pain, so a dictionary would tell you the difference in an instant. But, I would also assume you've done that, so your native language might not (I'm interested to find out what language that is).
An ache is a persistent discomfort, typically dull so that you can try to ignore it, but sometimes all-encompassing, yet not sharp enough to describe as pain. Your legs would ache after a tough run; you would not describe this as pain. You usually get a headache, not a head pain. You would suffer pain when you cut your finger, then experience an ache as the wound heals.
A pain is something more localised, often (but not always) short-lived, and something you'd be less able to ignore.
When you receive an injection, there is a pain as the needle goes in. During the following days, the surrounding area will ache.
There is considerable overlap between the two, and it would be quite acceptable to say "the ache in my shoulder is painful".
Poets and songwriters quite often speak of their heart aching. This fits well with a persistent sense of yearning or melancholy. If they said there was a pain in their heart, it would suggest a quite different emotion.
Describing pain and discomfort is difficult and subjective; I imagine in any language. When a doctor asks you how much something hurts, how can you explain in a reliable way?
Best Answer
The Oxford Dictionary gives both cipher and cypher in that order.
The preference for cipher over cypher seems to be a little greater in American English than in British English. The Corpus of Contemporary American English has just over three times as many records for cipher than it has for cypher. The British National Corpus has only one and a half times as many. It would not occur to me, as a speaker of British English, to use anything other than cypher.