I find it suspicious they look similar and both function to give explanations. Is it just coincidence or is ので kind of like the て form/mid-sentence form of のだ?
ので has similar meaning to から in that they can both mean "because, given that..."
のだ, or のです are versions of んだ/んです that are more often used in writing and they don't really mean "because", they just imply the speaker or writer is explaining or emphasizing things. It's quite hard to explain, you can read up on it here:
Yes, ので is related to のだ. This の sort of means 'thing' or '(the) thing is' in the sense of 'it is the case is that...'.
For this reason, the difference between さむい？ and さみいの？is that the first is a simple question, whereas the latter implies that you have reason to think that the other person is cold. Perhaps they are wearing a heavy jacket or they asked to put the heater on. Hence the second question is more like "is it that you are cold?".
んだ or のだ has a sense of explaining what the situation is. Hence you often here people starting with something like 今思ったんだけど so I was just thinking,... Another example might be スーパーに行くんだったら if you're going to the supermarket...(and it seems like you are)
ので is the て-form of this construction. It provides a sense of explanation, which is why it is similar to から and means something 'since' or 'because it is the case that'. This meaning is slightly different from から because in the construction AのでB I feel that it implies that the listener is aware of the circumstances or would agree with the statement A, for which B is a consequence.