Learn English – Climb a tree or climb over a tree

A sourcer is someone who collects a list of names and resumes from the internet.

A recruiter is someone who actually talks to the candidates after filtering the info gathered by sourcers.

Normally a recruiter will be a sourcer at the same time or was previously a sourcer.

"The dog was entangled in his chain" The dog was caught in the chain

"The dog was entangled with his chain" The dog and his chain were entangled, possibly in something else, possibly in one another - however, the latter makes no sense since the chain cannot be entangled in the dog.

"The dog was entangled on his chain" The dog was entangled in something, and positioned over his chain

In your example: "The dog climbed the fallen tree, then slipped and entangled his chain on/in/with it." It seems that you want to say that the chain got caught in the tree rather than entangled: "The dog climbed the fallen tree, slipped, and his chain got caught in the tree".