I agree with all of the above corrections, but as for Pyridines correction, I don't think "I learned it off by heart." really makes sense. It depends on what you are trying to say. You could say I know it by heart, which would mean you have memorized it to a degree that you will not easily forget it. Is this what you were trying to say?

My cellphone's battery is running out. Absolutely fine! We have run out of coffee = The coffee ran out.....NO We're out of coffee = We have run out of coffee. We ran out of coffee yesterday. I saw an article in a newspaper. This is fine! What is your question here? Is there any other more emphatic expression for "...by heart"? What do you want to emphasise? By heart = for memory. How are you supposed to emphasise it?

Add punctuation and these are all fine. There is a stupid joke about a fridge running that can be appalled ti your coffee and battery. "Well, you better catch it." is the punch line. Though it is fine as is; your battery is loosing power, has lost power. Though it is fine as is; We need more coffee, we are out of coffee. By heart = memorized = etched in your mind = unforgettable = remember like it was yesterday

"My cellphone's battery is running out" is broadly similar as "My cellphone's battery is about to run out", except that "about to" suggests that the "running out" will occur very soon. We have run out of coffee = The coffee ran out. Yes, that's the basic meaning, although to be grammatically consistent, you would say "the coffee has run out".... or, rather, "our coffee has run out" Is there any other more emphatic expression for "...by heart"? One similar, emphatic expression is "by rote"... although this suggests an element of compulsion, as in school children who are given texts to memorize in order to pass a test.

My cellphone's battery is running out: This is fine and means pretty much the same thing as '~is about to run out'. However I'd say that '~is about to run out' is slightly more imminent, the battery is very close to running out. 'We have run out of coffee' means the same as 'The coffee ran out.' You might be more likely to use the first one if someone has asked for coffee. Q: Could I have a cup of coffee? A: I'm sorry but we have run out of coffee. I saw an article in a newspaper: This sentence is fine. By heart: You could say that you memorised something but I think saying 'I learned it off by heart.' is slightly more emphatic. Hope that helps!