"So that is why a Trainer plane for instance (which has a flat bottom wing), will fly more stable and more slow - because it generates more lift in level flight?"

A flat bottom wing is a cambered airfoil, which happens to have the bottom surface flat, just for construction convenience.
As you say, this cambered airfoil has a high CL, generating lift at slower velocities; hence, it can fly slow enough for training.
Now, the stability of the model has nothing to do with the airfoil shape; the trainers are stable for other reasons.

"But for this same reason it will not fly or not fly inverted well because it will be "pushed" down?"

The cambered airfoil ("concave" shape) can fly inverted, but it needs to compensate for the "convex" shape with higher AOA and/or higher velocity.

"If you then have a symetrical wing, the plane will have "equal" lift inverted as up and therefore is good for aerobatics?"

Yes.

"But at low speeds and level flight it will not generate as much lift"

The wing must generate the same lift force, in order to equal the weight of the model.
For the same wing area, the symmetrical airfoil (lower CL), needs higher AOA and/or higher velocity than any cambered airfoil (higher CL).

"- therefore stalling earlier?"

Not necessarily. Normally cambered airfoils tend to stall at lower AOA's than symmetrical airfoils.
For any airfoil, the stall depends only on the maximum reacheable AOA, which is called critical AOA.