In my Artscroll siddur there is a prayer for the sick that one can add into "Refaenu" in Shmone Esre. I don’t for which kind of sick people to say it - that is, for people who are temporarily sick (pneumonia) or long term sick (cancer) or for those who have many health problems that they don’t seem to be able to get over? I have the same question about making a Misheberach at the Torah reading.
There are no rules, each one can pray as he wishes. Regarding Misheberach, one should not trouble the Tzibbur unless for a special temporary case. In Shmone Esre, each one can lengthen his own Tefilla as much as he wants. Rabbi Zalman Melamed