There are 2 different Corporate actions on TROV.
The first is a reverse split of 12:1. i.e. for every share 12 old shares held; will be converted to 1 new share. This is effective 1 June 2018. If held electronically, these will automatically get converted. Note there is no cash being given for fractional shares. i.e if an individual was holding 100 of old shares, these will be converted to 8 of new shares. There is no cash value for for balance fraction 4 shares. They are extinguished.
The second as linked, is a Follow on Public offering to raise additional capital. Individuals / Institutions can participate in this offering and get additional shares.
If a person is currently holding 1 share of TROV, are they exiting stockholders or new investors? What is the difference between the two?
The current shareholders are "existing". The "new investors" are people / institutions participating in the follow on public offering.
The book value is generally relevant during bankruptcy or other extreme events. For day to day; it does not matter to a large extent. The prospectus mentions that current book value as $0.9 [Assets - Liabilities]. Note this is not the price at which the stock is trading. It is more around $2.6. So the new shares will be offered near this price; so it would increase the book value of existing share holders as the new average book value will be around 1.76