Given a Total Cost equation $TC(Q) = a + bQ + cQ^2 + dQ^3$ what do the coefficients mean? For example $a$ is fixed costs, what are $b,c,d$ and how are they calculated?
There is no deeper meaning than what you already discuss.
The Total Cost function is always made up of fixed costs plus the variable costs (each of which may be zero). We have:
$TC(Q) = F \; + \, VC(Q)$
where $TC$ is the total cost function, $F$ are the fixed costs and $VC$ are the variable costs.
Fixed costs do not depend on the quantity, unlike variable costs. There is nothing more to it. In your case, $a$ is the fixed cost and the variable costs are $bQ + cQ^2 + dQ^3$. The relation between quantity and variable costs is somewhat complicated and non-linear in your example, but there's nothing wrong with that. The exact form of the cost function comes from the technical aspect of the production process, which may very well be complicated.
So $b, c, d \,$ are just exogenous parameters (imagine them as numbers), which give the shape of the cost function, but do not have any deeper meaning. Furthermore, they will not be solved for or calculated and are simply taken as given. This is always the case with parameters. In solving economic models, you take the parameters and exogenous variables as given and solve for the endogenous variables, which in your case probably involves the quantity $Q$.