D.B.
I think it's very hard to tell children how they are supposed to feel and how they are supposed to show that feeling.
Not wanting to cry in public makes sense. There's either a rush of people saying, "Don't cry" or a rush of people smothering and consoling. That's too much for a kid who may be on the shy side (especially based on your other posts about your daughter). I know you worry so much about her, and I wonder if the hovering is a problem for her sometimes.
A lot of kids laugh when they are nervous. I had a student who smiled when he was in trouble - it wasn't disrespectful actually, it was just an automatic response. There's also a more serious disorder that shows these behaviors, although I'm not suggesting he had it or your daughter does. We teachers were experienced enough to understand this, and we consulted with each other to see that it was across the board, and not a show of disrespect to a particular teacher. I think you should not worry about what a young child does in school ("What if...?") - this is elementary school and it's really not necessary for you to get so upset about this. Probably the teacher has observed this behavior. If you're desperately concerned, you can mention it at the parent-teacher conference or send a very brief note in at the beginning of the year. Don't go on and on - just say that you've noticed your daughter can laugh at inappropriate times as a defense mechanism, and for the teacher to let you know if it's a problem that needs to be discussed. Don't overdo this - teachers have many new students to deal with and get to know, and they don't need a big narrative, just a brief head's up.
Not knowing how to behave in a memorial service is totally age appropriate. Not wanting you to see her crying ("something in my eye") could be her way of protecting YOU from sadness. I think that's sweet, but I also think it's a concern if a child is trying to take care of the parent. I think kids learn by doing, and they learn by seeing. That includes ways in which to grieve - but young kids not yet missing the person who has died is so normal (because they don't yet realize what it's like to not have someone come back). Moreover, thinking about death and the loss forever of a loved one is very scary for children - can you imagine? Long explanations don't always have the effect we'd hope for - it's often too divorced from the actual event, or even at the same time, a kid doesn't see her own facial expressions and connect them to what you're talking about. A brief "we don't laugh out loud in church" should be enough to cue her to tone it down.
That said, you have expressed many concerns about your daughter over the years, and if you think this is part of a larger problem in many situations over many years, I think it's worth a discussion with her pediatrician and a referral to a family therapist to help you all through this with new coping strategies. If I look at all your posts not as individual ones but as different aspects of a large problem that has you extremely worried, it's a logical step to look at some neutral and professional input.