Behavior of different multiplied and added time complexities
Let $f(n)\in \theta(h_1(n))$ and $g(n)\in \theta(h_2(n))$.

We start with multiplication:
It is easy to prove (from the basic definitions) that $f(n)\cdot g(n)\in \theta ((h_1\cdot h_2)(n))$, and similarly for $\Omega,o,\omega$. 

Accordingly, multiplication of functions is preserved in all the asymptotic notations ($O,\Omega,\theta,o,\omega$) 

As for addition, the idea is to make the following observation: 
$$f(n)+g(n)=\max\{f(n),g(n)\}+\min\{f(n),g(n)\}$$
But $\min\{f(n),g(n)\}\le \max\{f(n),g(n)\}$, so $f(n)+g(n)\le 2\max\{f(n),g(n)\}$. Thus, we get
$$\max\{f(n),g(n)\}\le f(n)+g(n)\le 2\max\{f(n),g(n)\}$$

This implies that the asymptotic behavior of $f(n)+g(n)$ differs from that of $\max\{f(n),g(n)\}$ by a multiplicative constant, which means that they have the same order of magnitude, and in particular the relation is preserved under all the asymptotic notations.

To add some detail to the above, we use the observation that if e.g. $f(n)\in \Omega(h_1(n))$ and $g(n)\in \Omega(h_2(n))$, then $\max\{f(n),g(n)\}\in \Omega(\max\{h_1(n),h_2(n)\})$. This is relatively easy to prove.