Is it be correct to say that we live in a young (only 14 billion years old) universe? From what I have read, it seems that our universe is expected to function more or less as it does now for some $10^{12}$ years, possibly more. If that is correct, our universe's current age of $14$ billion years seems like this place we live in is still a baby, or at most a child.

I realize we are a long way from being sure about making such a broad characterization. Nevertheless, does it generally accord with current astronomical and cosmological thinking to say that our universe is still young?
An alternative way of looking at it is that we live in a dying universe. It may have a long, drawn out end, but the peak of star forming activity happened about 10 billion years ago, both in our own Galaxy (see Is there a stellar database that indicates how long ago stars in our Galaxy formed?) and in the universe as a whole (see Will new stars stop forming at some point of time? ) and most quasar activity petered out about 5 billion years ago (e.g. see  Why are all quasars so far away? ).

On the other hand, it is true that the number of (low-mass, very low luminosity) stars in the universe is still increasing (e.g. When will the number of stars be a maximum?), thanks to their very long lifetimes. So although the fireworks are over, the bonfire is still smouldering (to use a UK metaphor that will have others baffled!).

I guess it all depends on whether you are a glass half-full or half-empty kind of person.

Another thing to consider from a philosophical point of view is the Carter catastrophe. The basic idea here is that it is unlikely for us to be alive at a special time just after the beginning of the human race, and that this indicates we are shortly to be doomed. This is normally framed in terms of humans and our survival on this planet, but I suppose could also be applied to the universe as a whole. The arguments that the universe could last trillions of years are based on concordance cosmology with the parameters for dark matter and dark energy fixed. The problem is we don't know what dark energy is or whether it varies in time or exactly what the relationship is between dark energy density and the (negative) pressure it exerts. There are big rip scenarios that may see the end of the universe within a mere 100 billion years!