As usual it depends on your case. You can have a big immune response, but still within the range your body can take, and you won’t be feeling ill.

I think kids will often have very visible effects after vaccination, while adults in their prime are usually more solid and won’t notice much of what’s going on internally.

No. Feeling ill is mostly due to the innate immune response which declines with age (so seniors have fewer side effects). The adaptive immunity triggered does not seem to be correlated with the innate response or feeling ill.