Why does britney say Break the ice as “Break THE ice” not “Break THEE ice”? Doesn't “The” pronounce “Thee” if the words after it are plural?

The convention of pronouncing *the* as /ðə/ before a consonant sound but /ði/ before a vowel sound is not as mandatory as you might think; it's just common practice because it's usually easier to pronounce that way. It should not be equated with, for example, *a* vs. *an*, misuse of which *would* be considered an error (or slip of the tongue) in most circumstances.

No. The two pronunciations are basically interchangeable.

It's commonly done because it's easier for your mouth to move onto a vowel from another vowel sound, but it's 100% interchangeable. It's not a hard rule, as much as something done for convenience.

Where did you hear this nonsense? Not to mention, since when is "ice" considered plural? WTF is going on here?

The rule you're referring to doesn't actually exist (both pronunciations of "the" are interchangeable) and also "ice" is uncountable, not plural.

“Thee” is mostly just there to prevent an awkward glottal stop. Say it how you want.

But ice is not a plural word. Try it out with a plural verb after it and see what I mean. "The ice are cold." Nah. Also, I'm not sure the rule you cited is very reliable. I would generally just pronounce the word, "The" as something like "thuh" regardless of what comes after it. "Thuh" mountains, sure. "Thuh" single mountain, sure. Are you thinking of the older English (object-case) pronoun, "Thee"?