My advice is when the 16 month old is upset and wants attention when she can not have it her mother should go to her come her down by speaking to her. She can say something like "i know your upset and you want me to play with you but right now I can't." or "i know you are upset you have to go to bed but now we r going to sleep." this way the mother is acknowledging the babies feelings but still saying how it has to be. Ignoring the behavior I think is a big precursor for problems later. Even though she is very young she could grow up with the memories of being ignored when she was most upset which could lead to difficulty handling her emotions. After the mother goes and talks to her once she may start crying again and getting possibly to the point of vomiting but then her mother should go back to her and repeat the same thing. Example: "i know u r upset now but we r going to sleep now" after that if the baby continues which she very well may her mother should go to her and say something shorter like "we r going to bed now or good night" now the baby id not getting ignored but she is recieving less and less attention. It may take awhile but it is worth it and eventually her mother shouldn't have to deal with it at all. I am a firm believer that little kids often calm down when they knwo they r being understood. They don't have the words yet to explain so they communicate with there actions so like anyone else they want to knwo they are understood verses being ignored.
The reason I have thought so much about this is because when I was a child I would make myself pass-out and a doctor told my mom to ignore me when I did this. At this point I realize why I never told my mom about anytime I was upset I would rather cry alone if I was sad or angry becuase why tell her? She never understood before.
As for the 9 week old I think that it is more likely that the baby vomits when it is crying because sometimes babies do that they cry hard gag on their tears and spit up. 9 week olds really only know a few things they cry when they are hungry, need to be changed, are uncomfortable, tired or wnat to be held.