Those categories are somewhat irrelevant to Korean grammar. There s no equivalents to 'participles', and there are many different forms u could consider as gerunds (i.e. forms of verbs having an adverbial function). If u think of the gerund in the progressive, it would be the form in -고 (있다): 가고 있다 "be going", but it cannot modify nouns, and it is otherwise use for coordinating verbs together: V고 V고 "V and V and...".

Yes, there are equivalents. You you can say that they work like gerunds and participles. Instead of declensions, they are particle markers -(이)던 -는 -일. "Participles": (past) 달리던 사람 "person (who) was running"; (present) 달리는 사람 "person who is running" or "running person"; (future) 달릴 사람 "person who will be running". As for gerund, where verbs become nouns, one way is to express it is to add the phrase 이유로 "for the reason (of)" to the participle: (past) 달리던 이유로 "for the reason of having run"; (present) 달리는 이유로 "for the reason(sake) of running"; (past) 달릴 이유로 "for the reason of going to be running". Another way is to add -이기 to the verb: 달리기 (the act of running), it only exists in present form. Let's put it together in a sentence: 저달리는 사람은 달릴 이유로 달렸다. "That person who is running, ran with a purpose for running". It's okay to mix tense in Korean. Hope that helps.