From the CDC: Myocarditis and pericarditis after COVID-19 vaccination are rare. As of July 30, 2021, VAERS has received 1,249 reports of myocarditis or pericarditis among people ages 30 and younger who received COVID-19 vaccine. Most cases have been reported after mRNA COVID-19 vaccination (Pfizer-BioNTech or Moderna), particularly in male adolescents and young adults. Through follow-up, including medical record reviews, CDC and FDA have confirmed 716 reports of myocarditis or pericarditis. CDC and its partners are investigating these reports to assess whether there is a relationship to COVID-19 vaccination

There were 716 confirmed reports. Most clustered in a population of 18-25 year old males. There are about 9 million fully vaccinated 18-25 year old males. For that demographic the chance of getting this side effect is about 1 in 15,000 which is greater than 1 in 20000, the chance of an equally serious side effect that got Rotashield pulled from the market in 1999.

Does risk of myocarditis caused by  vaccine run in the family? So  if the parent gets  myocarditis from vaccine or natural covid; do the children  have an increased risk ? Other health risk factors are sometimes inherited, so it would be nice to know if vaccine side effect risks are too.

That way you could vaccinate  yourself and spouse first, and only then vaccinate the kids.

2)

Risks and side effects of vaccines: The US CDC lists several of them, but based on the list I don't see age-specific statements. What they say here is:

Cases of myocarditis (inflammation of the heart muscle) and pericarditis (inflammation of the outer lining of the heart) have been reported after Pfizer-BioNTech COVID-19 vaccination of children ages 12–17 years.  These reactions are rare; in one study, the risk of myocarditis after the second dose of Pfizer-BioNTech in the week following vaccination was around 54 cases per million doses administered to males ages 12–17 years. 

In general, adolescents ages 12 through 17 years have a higher risk for myocarditis than children ages 5 through 11 years. During clinical trials, no cases of myocarditis occurred in children ages 5 through 11 years who received the COVID-19 vaccine.

Okay, so the myocarditis risk seems to be below 54 cases per million (~0.005%). This is hard to judge because they don't mention the backgound risk in the population of this age group.

A German government website lists vaccine effects like headache: 76% and says this is from the study testing the vaccine for 12-15 year old teenagers. Well, maybe. The study for younger children lists effects as follows:

"Injection-site pain was the most common local reaction, occurring in 71 to 74% of BNT162b2 recipients. Severe injection-site pain after the first or second dose was reported in 0.6% of BNT162b2 recipients and in no placebo recipients. Fatigue and headache were the most frequently reported systemic events. Severe fatigue (0.9%), headache (0.3%), chills (0.1%), and muscle pain (0.1%) were also reported after the first or second dose of BNT162b2. Frequencies of fatigue, headache, and chills were similar among BNT162b2 and placebo recipients after the first dose and were more frequent among BNT162b2 recipients than among placebo recipients after the second dose. In general, systemic events were reported more often after the second dose of BNT162b2 than after the first dose. Fever occurred in 8.3% of BNT162b2 recipients after the first or second dose. Use of an antipyretic among BNT162b2 recipients was more frequent after the second dose than after the first dose. One BNT162b2 recipient had a temperature of 40.0°C (104°F) 2 days after the second dose; antipyretics were used, and the fever resolved the next day."

An illustration from the German newspaper Zeit.de (which says it is based on FDA data) compares control group and placebo group, and it seems that the risk for fever, fatigue, headache, chills, vomiting, pain, joint pain are indeed relevant. At the moment I'd say it all comes down to comparing relatively high probabilities of minor side effects to low probabilities of major effects of Covid. But note that the study is so small that it would overlook small-probability effects.

1)

I'll add some sentences on vaccinating children in the age group 5-11.

The US CDC recommends doing so. I'll first consider the COVID-19 risks and then hope to find more on the risks of vaccination. The CDC website recommending the vaccination this says:

"There are approximately 28 million children between the ages of 5 and 11 years old in the United States, and there have been nearly 2 million cases of COVID-19 within this age group during the pandemic. COVID-19 can make children very sick and cause children to be hospitalized. In some situations, the complications from infection can lead to death."

Here and in the rest of the cited website section, you will see words like "can" and "some situations", whereas in the section on vaccine safety, the words are "very rare". Nonetheless, at least some numbers are there:

"As of mid-October 2021, children ages 5 through 11 years have experienced more than 8,300 COVID-19 related hospitalizations and nearly 100 deaths from COVID-19. ...

Children who get infected with COVID-19 can also develop serious complications like multisystem inflammatory syndrome (MIS-C)— ... Since the pandemic began, more than 2,300 cases of MIS-C have been reported in children ages 5 through 11 years. Children with underlying medical conditions are more at risk for severe illness from COVID-19 compared with children without underlying medical conditions."

Based on this data, the risks can be calculated relative to cases; the resulting risk numbers constitute an upper bound for the risks (based on the data in the text). We cannot use this to calculate the risk relative to infections, because we don't know how many infections there have been; the upper bound is 28,000,000. We can thus calculate risks relative to a "population" of 2,000,000 as an upper bound or relative to a population of 28,000,000 as a lower bound, but note the words "more than" and "nearly" above. (In all of this I have to assume that the children were unvaccinated.) That said, the risks are:

• becoming a "case" given infection: At least 7.143%
• hospitalization: 0.03% - 0.415% (at least),
• MIS-C: 0.008%-0.115% (at least),
• death: 0-0.005% (at most)

(That said, the sentence "Children with underlying medical conditions are more at risk for severe illness from COVID-19 compared with children without underlying medical conditions." may motivate parents of children with underlying medical conditions, but also demotivate parents of children without such conditions.)

The Comirnaty study estimates efficacy to be 90.7% (95% CI: 67.7 to 98.3).

Now to determine the sensibility of a vaccine, I assume that these risks of Covid-19 should be compared to background risk? 

Relevant recent tweet by Divia Eden, see also the comments by David Manheim in thread:

Just a totally normal @TheLancet study that came out yesterday showing that 4.4% of kids experience Long COVID symptoms and this data was collected pre-Delta variant: https://www.thelancet.com/journals/lanchi/article/PIIS2352-4642(21)00198-X/fulltext 

3)

Finally, a comment on externalities. 

As far as I understand, the Comirnaty study does not include transmission-risk due to vaccinating children, and it cannot include that due to the study design. Nonetheless, the article includes the following sentences: 

"Direct benefits of preventing SARS-CoV-2 infection in children include protection against severe disease, hospitalizations, and severe or long-term complications, such as MIS-C. Indirect benefits include the likelihood of reduced transmission in the home and in school settings, including transmission affecting vulnerable persons, and safer in-person learning." 

The sentence about indirect benefits seems a bit misleading, however, because "preventing SARS-CoV-2 infection", as far as I understand, can seem quite different things. Transmission reduction due to vaccination in adult studies seems to be between 63 and 89 per cent, depending on the study and whether the data include the delta variant (I guess it's too early for Omicron data in this context). As far as I know, children are general less infectious than adults (due to smaller lungs), but that is mostly about short-term interactions. It is plausible that children that bring the infection to their parents also transmit the disease to them. So even smaller transmission-risk reductions would be valuable.

(As always, I'd be happy about corrections here.)

Interestingly, Germany's Standing Committee on Vaccination (StiKo) does not yet recommend vaccination for 5-11 year olds due to lack of data (I can understand that point) but at the same time says:

It is also recommended to vaccinate children around whom there are contact persons at high risk of a severe course of COVID-19 who cannot be protected or only inadequately protected by a vaccination (e.g. the very old and immunosuppressed). 

I don't know whether there is data for that which I have overlooked. Moreover, the StiKo's recommendation is said to be based on the consideration that it is unjust to let children be vaccinated just to protect others (in particular adults who decide not to get vaccinated just because they do not want to). However, it is unclear to me whether the StiKo's position is that you should not count contributing to public goods at all, or whether the cost is just too high to justify that. (The StiKo recommendation is not yet published in the Robert Koch Institute's Epidemiologisches Bulletin.) In any case, I would consider that the group for you provide a public good can sometimes be small enough to be very concrete - for example, I guess that if 20 kids in the kindergarten group are all vaccinated, the risk that the kindergarten has to close for a week because of an infection will be much lower.