Usually one defines an action of a topological group $G$ on a topological space $X$ as a group action of an abstract group $G$ on the set $X$, such that $\cdot: G\times X \to X, (g,x) \mapsto g\cdot x$ is continuous. This implies that $\rho: G \to \text{Homeo}(X), g \mapsto (x \mapsto g\cdot x)$ is a group homomorphism.
Why do we need continuity of $\cdot: G\times X \to X, (g,x) \mapsto g\cdot x$? Only to get maybe stronger theorems?
One could also consider an abstract group $G$ and an abstract group action on the space $X$ and just require $x \mapsto g\cdot x$ to be continuous for all $g \in G$ (i.e. above definition but topology of $G$ forced to be discrete). The topology of the quotient space is independent of the topology of $G$?
To address your second paragraph, there is indeed a whole theory of continuous group actions, and an even more specialized theory of smooth group actions, and a still more specialized theory of Lie groups actions. And there are applications of all of these theories. I would not say that the theorems are stronger, as you suggest; I would simply say that they are different. I suggest, if you are interested, that you simply pick up a good book on topological groups or on Lie groups, and start reading.
In your third paragraph, the topology of the quotient space is, by definition, the quotient topology defined by the decomposition of $X$ into $G$-orbits. This decomposition is well-defined independent of the choice of a topology on $G$, and so yes, the topology on the quotient space, also called the orbit space, is independent of the choice of a topology on $G$. Nonetheless, in the more specialized theories there are more specialized kinds of quotients. See this discussion of orbit spaces of Lie group actions, for example.