Why does every member $f(n) \in \Theta(g(n))$, and $g(n)$ have to be asymptotically non-negative?
Mostly for convenience's sake.  Since big-theta notation specifically tends to be used mostly for algorithmics (as opposed to $O()$ and especially $o()$, which make occasional appearances in more 'traditional' mathematics for talking about goodness of estimates) and is generally used to measure 'usage' quantities of some sort (e.g. CPU cycles, memory, comparisons, etc) there's generally little lost by requiring the quantities involved to be non-negative, and it simplifies some of the definitions and results.

Note that unlike with $\Theta()$, the definitions of $O()$ and $o()$ are often phrased in terms of absolute value, making this point moot - can you see why that idea doesn't make sense for $\Theta()$?