Perpetual Youth Model
I haven't read the paper in full detail (I should, it's interesting) but let me give you my interpretation of the paper's title.

Blanchard uses a constant probability of instanteneous death for each agent. As mentioned in footnote one of the paper this is most realistic as assumption for people between 20 and 40. Agents in the model therefore are like young people for their entire life, hence the title.

As for your question 2, the way I read the paper yes there are many people alive of all ages. After all at each instant in time a new cohort of size p is born (and starts dieing off immediately at rate 1/p). So eventually at time is infinity that cohort will have completely passed away.