It does seem really weird to me that a resident would get chewed out over not attending the welcome BBQ. My residency puts on a welcome picnic too, but there's no pressure to attend, because the people at the residency I attended are laid-back, normal human beings instead of control freaks with sticks up their butts.
The way I see it, if you want to make it clear that everyone has to attend, don't call it a BBQ. Call it a mandatory meeting that just happens to have lunch served.
Of course, I am the kind of person who has always disliked corporate team building and other types of Mandatory Fun.
I am sorry to hear that it sounds like your family is in chaos right when things are so stressful for you. I definitely think some residencies are more understanding about this type of thing than others are, and I can imagine that having these worries about your parents is probably amplifying all the other stress by magnitudes. What was the reason you had to apologize for taking a vacation day though? Was it during a rotation where vacation isn't normally allowed? Personally I think residencies should not be service driven and should be able to function without relying on the residents, since life does happen to people and sometimes illness/tragedies/etc. mean people really cannot be at work. Unfortunately the reality is that sometimes a resident being gone does cause massive issues on rotations where they have come to rely on them, so people do resent it even if you have a good reason for being gone.
Truthfully, there was some resentment in my residency program when one of our residents took maternity leave, because it did have a negative effect on the rest of us, even though we all understood that of course she was entitled to take time off for her child.
Now, I agree with NickNaylor that the punctuality thing is a legitimate concern. You need to try to break that pattern/perception of you ASAP. Perception is far more important than reality in residency, and you do not want to be perceived as a slacker who always shows up late or (far worse) doesn't show up at all. Truth is, getting through residency is really not about your knowledge base or how much your patients like you or how much you actually do help patients. Getting through residency is about being perceived by the people who matter as being likable, responsible, hardworking and reliable.
This means the little things are crucial: Answer pages right away. Do your notes on time. Show up for things when you're supposed to be there.
Even if you have to set all your clocks 20 minutes ahead of schedule to make sure you aren't late, do it. Even if you think you have a good reason/excuse, if it becomes something you're "Known For" (faculty definitely talk amongst themselves about residents) then it will hurt you badly.