1) There is a difference between 冷得 and 冷到. While 到 has the meanings of "to", "up to", "until (a time)", and "arrive"; 得 simply means "obtain" or "permit". It is clear that 到 has the implication of an event related to the change in space, extent, or time, which is lacking from the meanings of 得.
Sometimes the difference is not that noticeable, such as replacing 得 with 到 in the lyric 冷得讓我無法多愛一天, however, let's expand the resulting sentence 冷到"一個程度"讓我無法多愛一天, now we can clearly see the difference - the expanded sentence can't be replaced by change 到 to 得.
For the lyric, I wouldn't try to attach a meaning to 得, as it is rather meaningless, and redundant, as 冷讓我無法多愛一天 is a good matching sentence as well, but for better rhyming and fitting the tune, a word is thus needed. In general conversation, I would say 冷"到"讓我無法"再"多愛一天 is more suitable in expressing one's regret and sadness.
2.) 我唱得她心醉 我唱得她心碎. Rather than word-to-word translation, let me apply the same technique above - expand the lyric to its fill context as it will be in writing.
我唱得"讓她感到"心醉 我唱得"讓她感到"她心碎. In here, 我唱 = my singing, these sentences are general expressions that do not have a sense of time, they describe a timeless event.
In contrast, 我唱到她心醉 我唱到她心碎 is exactly what you have translated, there is the implication of time and a sense of the continuation of the action, so until (直到) is a suitable conjunction to add to the translation.
The lesson here is while confused about the proper/correct meaning of two Chinese sentences, it helps to expand the often shortened sentences to their full context as in writing, and the answer will be much clear, it especially applies to lyrics and modern poems.