Using two "te" form verbs together


Heard this in Yakuza 6 (Japanese video game) and was wondering if anyone would
be able to explain the nuance in using of the two "te" form verbs.

> 先生に 話 きて いて

I added the spaces to separate the words I am interested in. I understand the
base of what the phrase means, I just want to understand the grammar.

  1. Are those the "te" forms of "くる" and "いる"?

  2. Does it mean "I went and had a talk with the doctor" or "I had a talk with the doctor" or "I was having a talk with the doctor"?

