1) 스로 is not used by itself nor very common in other words. It has the feel of "in such a way".
I can only think of 스스로 (on one's own; of one's own accord) that has 스로 in it.
2) First, 되는 is "becoming" grammatically, but you can also think of it as 하다.
~나 as a particle has two major meanings. The first one is when enumerating things (이거나 그거나 저거나 다 같은 거야), and the second is to emphasize a quality(quantity, size, etc) of something (이 만큼이나 했어? vs 이 만큼 했어; 세 번이나 시도했는데 실패했다).
In this context, it's used in the second sense. It is expressing "as if it is something very serious".
3) The literal meaning is more like "threw a word like it's nothing", as 툭 describes something light and insignificant, something nobody would even notice, never anything big like a "thud".
Also, "툭하면 XX 한다" is a common expression for "Do XX habitually (for no particular reason)".
4) Yes, 새삼 is from 새삼스럽다. It's an adverb while 새삼스럽다 is an adjective.
새삼 and 새삼스럽다 are much more nuanced than simple "again" (there is 다시, 또, 거듭 for "again").
It describes the feeling you get when you notice in a fresh new way something you have forgotten about or taken for granted for a while. It has a rich meaning and nuance for which I can't think of any matching word in English.
5) You can remember 들여다보다 just as a unit word. It's never broken apart in real usage.
As for the derivation, it is probably like 들이다 + 아/어 + 다가(first doing something, and then) + 보다, but that's just my guess.