Generally speaking, I'd say if you use it for one reference type, you should use it consistently for others as well. The best way of justifying it would be that when you give a specific number to an object, "Equation (number)" becomes the title of that object, whereas something like "the aforementioned equation" is a more general term.
In my field "Theorem", "Property" and "Lemma" aren't usually things that are referred to as such in writing (pretty sure this just depends on which subject area you're writing for), but certainly if e.g. "Section 3" and "Figure 5(a)" are capitalised, so too should "Equation 2.3".
I also see many cases where such references are abbreviated to "Fig.", "Sec.", "Eq.", etc if they are in the middle of a sentence and left as "Figure", "Section", "Equation" if at the start of one - but as far as I can tell, this is just a matter of personal taste unless a specific style has been demanded by a journal. Just make sure that whichever one you use, you remain consistent throughout the text.
(as an aside: In your first example,
As seen in last theorem would become
As seen in the last theorem if anything; but
As seen in Theorem 1 would generally be preferable to avoid any ambiguities)