Must, like all modals, takes the base form of the following verb, not a to infinitive. Always.
Must is also a noun, meaning "something that you must have". Your first sentence needs "a", and says "Languages are a must to learn", or to turn it round "To learn languages is a must". Even with this nominal use of must, the ...
I think you're being misled by the fact that in some circumstances could functions as a past.
Historically, could was the past of can (and might was the past of may, incidentally). This still appears when we back-shift for indirect quotation:
He said "I can do it" -> He said he could do it.
But in modern English, as you note, could has other ...
a) To apply for the job, the candidate must have been born on or before 1st Jan. 2000.
b) To apply for the job, the candidate has to have been born on or before 1st Jan. 2000.
Correct, although (a) is perhaps easier to read.
c) To apply for the job, the candidate had to have been born on or before 1st Jan. 2000.
Wrong: the tense implies that it ...
"Will" is used to talk about the future: "we will go tomorrow".
"Would" is used for (a) the future in the past and (b) the conditional.
1: "We will go tomorrow."
2a: "I said that we would go tomorrow"; "I knew that we would go"; "I thought that we would go"; "I had already decided ...
Modern English, apart from a few vestiges, does not express mood by morphology: instead, it has a range of modals (such as will, can, must, should) each of which has its own range of meanings.
Some of them can be used to translate particular morphological constructions in other languages: but in my view nothing is gained by trying to put labels on the ...