The contents of all vaccines are published openly and in great detail. The exact proportions of the different ingredients and how they are formulated are not openly published, because they represent the IP of the company, but are reported to each regulator where some sort of licence is sought, e.g. CDC/FDA, MHRA, EMA etc. The EMA is made up of representatives from the regulatory bodies in each EU member state so each of those regulators also knows the detailed formulation. Every batch is tested and the results reported to the relevant regulator.
As an example this article
details the full contents of the Pfizer-BioNTech vaccine in Paragraph 1. The article also gives masses of information on the observed side effects. In the introductory paragraph it references a pdf fact sheet that is issued to every centre administering the vaccine so they are alerted to them. The submission for regulatory approval also gave details on the relative occurrence in the trials of each of these side effects. There is a yellow card system through which any medical practitioner or any individual can report side effects that they observed/experienced so that area is constantly evolving. It is completely normal for rare side effects, which didn't appear in the trials, to be reported when the vaccine is rolled out to the whole population. Similar information is widely available for all vaccines that have gained some sort of approval.
The advice on who should receive the vaccine is based completely on balance of risk. Mild and moderate side effects are largely ignored, although it is important that people receiving the vaccine are informed about them so they know what might happen to them. Serious side effects are evaluated further and can result in changes to the administering advice. As an example this article
describes the EMA review of the blood clotting issue possibly associated with the AstraZeneca vaccine.
Assessing the risk/benefit balance is complex. The risk of developing these rare blood clots is extremely low in all age groups and the risk of dying from them lower still. The risk does seem to be slightly higher in younger people compared to older people. That on its own would not rule out giving the vaccine to younger people but younger people are much less likely to suffer serious illness and death if they catch COVID so the risk/benefit ratio is less favourable. On the other hand young people spread the virus more rapidly and it does eventually spread into older age groups where it is more likely to cause serious illness and death. It is these sorts of complex interactions mixed with political considerations that can result in different governments reaching different decisions.
Some hard facts, with a little bit of assumption. So far there have been a little over 163 million recorded cases of COVID world wide. We know that about 30 % of people who catch COVID have no symptoms and another 30 % have mild or moderate symptoms. Let's assume that none of them get tested so the recorded cases only represent 40 % of people who have caught it. That implies that a little over 400 million people have actually had COVID. Worldwide there have been 3.385 million deaths from COVID so that represents a Case Fatality Rate of about 0.8%. Lockdown measures do not stop the spread of COVID, they simply slow it down so that healthcare systems can cope so, in the absence of a vaccination programme, eventually the whole population of the world would catch it, i.e. 7.7 billion people. With a CFR of 0.8 % that would result in 60 million deaths.