In this case, "도경" is a person's name. Also, "도경" can definitely refer to "provincial police" ("-도" or "-do" is how different provinces are expressed in Korean, just as in Gyeonggi-do, Gangwon-do, or "도지사" - pronounced "dojisa" meaning a provincial governer), it does not mean a "policeman" or "policemen". It is used to refer to police headquarters located in each province rather than individuals in police force.
From the given sentences, especially the second one, it could be assumed that 도경 is sleeping or at least trying to but clearly having hard time doing so. He is very alert, edgy and sensitive to what is happening around him asevident from him "waking up to even the slightest noises of tiny animals. Therefore "소스라치다" and "까무룩하다" here basically show that 도경 is falling into a light sleep only to suddently wake up, shuddering or experiencing hypnic jerk (you know, one of those falling sensations that trigger your reflex, causing you to wake up suddenly and leaving you with a rather unpleasant feeling). As mentioned already, the second sentence is basically saying that 도경 gets woken up by even the slightest noises or by "수" (another person in the story, I presume)'s hand slipping out. Hope this helps.
In fact, you are just about right with your translation. Of course, like you said, it is not the exact equivalent in a literal sense, but you do seem to have a more or less clear idea on what "까무룩하다" means. Just to clarify, "까무룩" itself is a native/indigenous Korean adverb describing how one's consciousness/state of mind is at a given moment (rather than one's mental "system" which could connote a rather permanent state of mind of an individual - and sounds rather unnatural to begin with). The closest English equivalent would be one's consciousness being "blurred/blurry" whether due to fatigue, drowsiness, etc. You can easily find a similar expression, "가물가물" or "가물가물하다", which refers to hazy, dizzy, blurry consciousness (or memory or state of mind or whatever is being described). Personally, I'd say it is much more commonly used than "까무룩". In any case, think of it as describing flickering lights, just like Do-Kyeong's (who seems rather stressed and fatigued) consciousness.
As I've mentioned, "소스라치다" here refers to Do-Kyeong having that hypnic jerk, jumping at and waking up to even the slightest noise. Your translation of "반복하다" is spot-on, so assuming there aren't any underlying significances, the sentence is basically saying that Do-Kyeong's is repeating the very actions specified: "falling into a light sleep" and "waking up to even the slightest noise, experiencing that unpleasant hypnic jerk, probably ).
Just a friendly reminder that "Do-Kyeong" is his/her name, not "Do Kyeong-Eun". As you may already know, "은" is a subjet and topic marker, not the part of his/her name. Hope this helps.