Hello *Louis*,
As you may know "펑" [P'ŏng] or "Peong" is a commonly used onomatopoeia for describing an explosion (be it small or big) or a sudden pop as is the case here. "-하다" can be understood as "to make ~ sound" and in fact is often used in conjunction with different onomatopoeic expressions: 꽝 하다, 쿵 하다, 펑 하다, 엉엉 하다. It is easier to understand the whole as a verb in itself: for instance, "쿵 했다" (to thud).
"눌려 담겨있다" or "눌러져 담겨있다" does have two verbs combined: 눌러지다 (to be squeezed) and 담겨지다 (to be put into something). You may come across this kind of compound verbs quite frequently: for example, 잘려 담겨있다, 씻어 담겨있다, 당겨 벌려지다, etc. In any case, "눌려 담겨있다" could be understood as "to be crammed into/jammed into/squeezed into".
When persistently opened a lid dented that doesn't open easily, Golden yellow faded brush heads that were squashed in a narrow tin unfolded with a pop like blooming flowers.
Just a few pointers here and there for a better translation: the sentence sounds more natural if translated as follows:
"when (I/he/she/whoever in question) forced open the lid dented and stuck tight, yellow faded toothbrush heads that were squeezed/jammed/squashed into it popped unfolded like a blooming flower/popping flower pod."
The use of the adverb "persistently" is unnatural here. "Persistently" is not the ideal choice for describing actions, such as opening or closing as "persistently open(ed)" is more often than not used to describe something that is open all the time or for a long period of time. "우겨 열다/억지로 열다" is better translated as "to force open". "Persistently opened a lid...that doesn't open easily" just sound redundant so it is just better to opt for "stuck tight", "closed tight", "sealed tight", etc. (Also, taking into account how toothbrush heads are "누렇게" discolored and I am assuming, by all means, not in a visually pleasant way

, it'd be better to just say "yellow", "brown", "brownish yellow" or even "shades of yellow", just like how you'd describe discolored teeth or 누런 이빨.)
Hope this helps.