Wolfsheim is a gangster. It is implied throughout the novel that whatever business dealings Gatsby is involved in (legal or otherwise), Wolfsheim is his backer. During Chapter 4, Wolfsheim calls Gatsby a "perfect gentleman" and the kind of man you would "introduce to your mother and sister." Gatsby is the perfect front for someone like Wolfsheim. He has a nebulous background, no family ties, has personal attractiveness, and most importantly, is naive. Gatsby really fails to see people for what they really are. When Nick questions why Wolfsheim wasn't arrested for fixing the 1919 World Series, he states that "They can't get him, old sport. He's a smart man." So, when Gatsby is murdered, Wolfsheim cuts all ties with him. His excuse to Nick that we should "learn to show our friendship for a man when he is alive" is a pure crock. Wolfsheim used Gatsby for his attractiveness and gullibility. With Gatsby gone, Wolfsheim crawls back to where he came from--under a rock.
When Nick Carraway is trying to get people to attend Gatsby's funeral, one of the people he tries to get is Meyer Wolfsheim. He calls Wolfsheim's house a bunch of times and gets no answer. Then he sends his butler over with a letter and gets a note in response.
The note says that Wolfsheim is too busy to attend. It says he is tied up with some important business. It seems to me that maybe Wolfsheim just didn't want to draw attention to himself because of being involved in organized crime.
After the letter, Nick actually went to go see him. Wolfsheim used to attend funerals of those he worked with when he was young:
When a man gets killed I never like to get mixed up in it in any way. I keep out. When I was a young man it was different - if a friend of mine died, no matter how, I stuck with him to the bitter end.
This shows Wolfsheim was indeed familiar with death, but has hardened himself. My guess is that Wolfsheim is in a business where people die and to become sensitive to every death would mean regular pain and grief.
Concerning your question about The Great Gatsby, not much is said in the novel about Meyer Wolfshiem's refusal to go to Gatsby's funeral.
He tells Nick that he would "like to come." He also says that it was different when he was younger, that when he was younger if a friend died he would stick to him to "the bitter end." And he does seem fond of Gatsby. But he says, "I can't do it--I can't get mixed up in it." And he doesn't.
That's what is revealed in the novel. Anything else is inference and conjecture. That said, the feeling seems to be that Wolfshiem cannot afford to be publicly connected to Gatsby. Going to Gatsby's funeral, it would seem, might draw attention to him and increase his criminal profile.