Fixed cost of a firm
It is the first one, $TC(0) = FC$. This is the definition. 

Also consider that it is not clear what is "transformed by $q$ in some way". In case of
$$
\frac{5q}{q+1} + \frac{5}{q+1}
$$
are the two fractions transformed by $q$, or should I just sum them up to 5?

With your function, one can rearrange it to
$$
TC(q) = \frac{5}{q+1} + 5 + 5q + q^2 = -\frac{5q}{q+1} + 10 + 5q + q^2.
$$