In the 'Latest Reputation Received' thingy, the icon next to the thread name sometimes shows a green and gray/black sometimes. What is that supposed to mean?

How much reputation that particular post has received (green = good, grey = neutral, red = bad). Click on the 'Add to X's Reputation' to launch a dropdown menu showing the reputation received.

Ok. I got a really helpful reply from a member. How do I make sure that he gets the 'green' against his name?

Select "I approve" to the respective post (and write some comments if you want to) and click on "Add to reputation".

Reputation, good and bad, means zero here in the Coffee House.

Not really ... it doesn't count towards the poster's numerical reputation or rep power but it still counts towards the little green/red rep bar within the post, and the comments are still visible both within the posts as well as in the poster's member profile.