No, in statements prefaced by the expression “I didn’t know/think…”, I don’t think it’s advisable to always use the past tense for the verb in the relative clause. The present tense is mandatory when stating an absolute or universal truth, of course, as in “I didn’t know that it’s the Earth that revolves around the sun and not the other way around, until my science teacher told us in class,” but it’s not the only situation that requires the present tense rather than the past tense for the verb in the relative clause. We have to decide that on a case to case basis.
To begin with, take the first sentence you presented: “I didn’t know Lady Gaga could sing well.” The “could” in that sentence actually isn’t the past tense of the modal verb “can”; it’s simply the form of “can” that expresses ability of a general, indeterminate degree (in that sentence, it’s the adverb “well” that indicates a measure of that ability). When you talk now of something that you didn’t know sometime in the past, the form to use is obviously “could.” But if you are talking right now of something you realized right before the moment of speaking, you can use either the modal “can” or the present tense of the verb: “I didn’t know Lady Gaga can sing well” or “I didn’t know Lady Gaga sings well.”
The same thing applies to the second sentence you presented: “I didn’t know you drew/could draw that well.” If you realized that person’s ability to draw just now, right before the moment of speaking, you can tell that person “I didn’t know you can draw that well” or “I didn’t know you draw that well.” The reason a lot of people use “could” as default usage is this: very often, the realization they are talking about took quite some time in coming, not a realization that dawned on them right before the moment of speaking. It therefore shouldn’t be used as a blanket prescription for the usage of “could” for all statements prefaced by “I didn’t know/think…”
It’s also important to keep in mind that “can” is stronger than the modal “could” but weaker than the present tense of the verb in expressing ability or possibility. It’s therefore perfectly OK for you to make the following statements about a realization right before the moment of speaking: “I didn’t know you can draw that well.” I didn’t know you draw that well.” But if the drawing ability of the person you are addressing doesn’t particularly impress you, you can just say “I didn’t know you could draw.” Without any qualifier like “well,” that’s actually the weak form of the present-tense modal “can,” not the past-tense modal “could.”
As to doing research on the proper usage of “can” and “could” by watching episodes of English-subbed animè movies on DVD, I don’t think that’s advisable at all. I can tell you very frankly that those written translations of foreign-language dialogue are sometimes so laughably inaccurate either from a grammar or semantic standpoint. In fact, my two grown-up sons—both sensitive to English misuse like me—often heckle those translations for their looseness. We get lots of fun together every time that happens, which I must say is very often for comfort.