No, it's not right. I think the distinction is that "slime" is never used to refer to anything produced by humans. Snails make slime, earthworms are slimy, frogs are slimy, lettuce that has gone bad in the refrigerator is slimy. Stuff in the throat is "phlegm," "ph" pronounced "f" as usual, "g" not pronounced. To me it is both the formal and informal word; I can't think of any humble Anglo-Saxon word for it. To me "snot," the informal and slightly vulgar word, only refers to nasal mucus. "Slime" always means something disgusting or offensive. You might describe a politician as "slimy," meaning "he is like some vile, verminous, crawling snake or earthworm." (In real life snakes are dry, but most people think they are slimy because they look glossy.) There is a bulk-forming dietary supplement (brand name "Metamucil") that mixes in water to produce a thick, viscous drink; it would be natural to say "I can't stand Metamucil, it's just too slimy." Phlegm was one of the "four humors" of pre-scientific medicine, and there are terms for four different personality types supposedly caused by them. The terms have persisted, and--at least in old novels!--you may hear people described as "sanguine" (blood), "choleric" (yellow bile), "melancholy" (black bile), or "phlegmatic." The "g" in "phlegmatic" is pronounced, and it refers to someone who is relaxed and easy-going.

In English, phlem and mucus are called "snot". This is what you would say was in your throat.

"Slime" is a similar, a green and unpleasantly viscous substance found on some rocks, or under snails. It's also a gag (prank/joke) toy popular with children and practical jokers.

Yes, "slime" is not the right word here. We would have to clarify. You might have been using "slime" as a metaphor for a disgusting drink. "to have phlegm" is the most natural option for me.

No, you would not use the word slime to talk about something in your throat.