Learn Farsi: من احساس این نقش را دقیقاً درک می‌کنم. - I understand the emotion of this role exactly.

The word râ marks the entire specific direct object. Here, the object is the whole phrase ehsâs-e in naghsh (the emotion of this role). Therefore, râ goes at the very end of that entire noun phrase.

Could I use mifahmam instead of dark mi-konam here to mean 'I understand'?

Yes, but dark kardan carries a slightly deeper nuance. While fahmidan is great for facts or general comprehension, dark kardan means to deeply comprehend, perceive, or empathize with something. This fits perfectly when talking about feeling the emotions of an acting role.

Where is the hidden 'ezafe' (the linking -e sound) in this sentence?

It appears between the emotion and the role. You pronounce the phrase as ehsâs-e in naghsh. The -e connects the noun ehsâs (emotion) to the specific thing it belongs to.

Why does the word daghighan end with an 'an' sound?

This is a common way to form adverbs in Farsi using a grammar rule borrowed from Arabic. Adding the -an suffix (written with a double accent mark called a tanvin) to an adjective like daghigh (exact or accurate) turns it into the adverb daghighan (exactly).