Asking “What do you do for a living” has the potential for a lot of problems. But chopping off those last three words removes a lot of the potential problems, and opens up the possibility of a lot more interesting answers. Even if there’s no awkwardness, “What do you do for a living?” might just get you “I’m retired”, while “What do you do?” might get you “I volunteer down at the local wildflower garden”, or the like.
And “How do you know <host>?” is almost guaranteed to be safe, but it also doesn’t lead into further conversation nearly as easily.
A true classist would ask "Where do you Summer?"or “So, how are you connected to the Host/Hostess?” or “Where did you go to school?”
The most inveterate snob I ever knew was addicted to the second, and would then proceed (if you let him) to nose out every life connection you had ever made and categorize you accordingly. If you didn’t let him, then you went in the box with the “I don’t know, but clearly raised well” crowd.
“How do you know the Host” can also be asked innocently though, and is not a bad starter as starters go.
I’m not against the job question, but it seems to be quite a difficult one for stay-at-home parents. I don’t know why they feel badly about answering but they always seem to. In this area, (Wash. DC) there are a larger than usual number of people who cannot discuss their jobs, so it’s really de rigeur to change the subject quickly if anyone hems and haws.
Another not bad one is “What do you do for fun?” though for much of my adult life the answer has been “Listen to the radio on my way home from work.” One time in particular it felt like a sort of third degree which lead to a serious reckoning of my job and right to an actual life outside of it. But I’m sure the man asking was just hoping I might like to see a movie some time.
A good way to avoid awkwardness is to begin with a vague statement about yourself, like “I really can’t recall the last time I went to a BBQ like this. Do you get to often?” or “You know I used to love to make curries like this one, do you cook?” There is always something at hand one can comment/invite comment upon.
A lot of this social thing is about taking up the thread of conversation and allowing small talk to be small talk. While it’s worthwhile to examine the accepted forms every so often to see how they measure up to modern standards, looking for offense in pat questions also tells the questioner everything they need to know about you. If your conversational partner is not taking up their part, well, there you are.
The couple that just lost a child should not be at a party. They should be out looking for their lost child.