I'm pretty sure about it but I'll ask.

Runner ecounters Chum during the run and doesn't break subroutines, he proceed and ecounters Data Mine, not breaking subroutine as well, does he take 1 dms from Data Mine and 3 from chum, or only from Data Mine?

RavenLord said:

Since you already know the answer, I just want to ask: why wouldn't he take the damage from Chum?

my friend, after he read chum+woodcutter rulin, suggested that chum won't trigger because of lack od ICE (which was Data Mine)

The 2 situations are completely different: Data Mine gets trashed because its routine executes. So there was a routine that was left unbroken. With Woodcutter, there is no routine left unbroken, because there was no routine in the first place.