Are you willing to get the Covid vaccine when offered?

It would seem unlikely. Your body would still have all sorts of antibodies present. Of course, there might be the possibility of having something like Delta and then being exposed to Omicron....I'm not sure if that would cause an issue.
From what I understand, It would. So the main mechanism of infection is the spike protein. The spike protein is so different with Omicron as compared to Delta or any previous variant that it's certainly possible to catch both back to back. Omicron is in the process of overtaking delta though, so the hope is that it just drive it to extinction and delta is by and large a thing of the past.
"Those who have been fully vaccinated and around someone who has COVID-19 are recommended to get tested between five and seven days after their exposure, according to the most recent guidance from the CDC, though previous guidance had been between three and five days."

OK so if your child has Covid, what is "five and seven days after their exposure,"? I'm getting exposed every day.
It's 3-5 after your first known exposure. So if your kid test positive on monday, if you're concerned you would want to test thursday and/or friday depending on how you look at the days for yourself. You could also do both if it wasn't prohibitively expensive for you. The longest you would wait would be Sunday. If it was me, I would probably test once on Thursday (in this scenario), then if I had a negative but developed known symptoms test again as soon as I could as those symptoms developed.