Capitalization of theorem names
In English, proper nouns are capitalized. The numbered instances you mention are all usages as proper nouns, but merely refering to a lemma or corollary not by its name is not using a proper noun, and so is uncapitalized.
Thus, for example, one should write about the lemma before Theorem 1.2 having a proof similar to Lemma 5, while the main corollary of Section 2 does not.
Edit. Well, I've become conflicted. The Chicago Manual of Style, which I have always taken as my guide in such matters, asserts in item 7.136 that "the word chapter is lowercased and spelled out in text". And in 7.141 they favor act 3 and scene 5 in words denoting parts of poems and plays. This would seem to speak against Section 2 and possibly against Theorem 1.2. In 7.135 they say that common titles such as foreward, preface, introduction, contents, etc. are lowercased, as in "Allan Nevins wrote the foreward to...". This may also be evidence against Theorem 1.2. But in 7.147 they favor Piano Sonata no. 2, which may be evidence in favor of Theorem 1.2. But they don't treat mathematical writing explicitly, and now I am less sure of what I have always believed, above. I do note that the CSM text itself refers to "fig. 1.2" and "figure 9.3", and not Figure 1.2, which would clearly speak against Theorem 1.2. So I am afraid that I may have to change my mind about this.
No. "Theorem 2.4" is a title, hence capitalized. But the word "lemma" in "the previous lemma" is simply a non-proper noun, hence uncapitalized in English. In both your examples I would use the second version.
Although I wrote the above paragraph as though it were definitive, it is of course only my own opinion. There's (almost) no such thing as universally accepted usage.
Also, although I would write "The proof is postponed to Section 4," I'm not too bothered by "... to section 4." I have no good reason for being less rigid about this than about theorem capitalization. I've also been known to be inconsistent about things like "the Hahn-Banach theorem".
Donald Knuth et. al. give a clear rule for capitalization in Mathematical Writing:
19. Capitalize names like Theorem 1, Lemma 2, Algorithm 3, Method 4.