Ranges of operators acting on Fréchet Spaces

Question

My question is in the spirit of Reference request: Baire's theorem for operator ranges. It mentioned that :

1. Finite intersections and sums of operator ranges are operator ranges.

2. Images and pre-images of operator ranges under bounded linear operators are, again, operator ranges.

3. If $F$ is the algebraically direct sum of finitely many operator ranges $U_1, \dots, U_n$, then all these operator ranges are closed (and hence, the sum is also topologically direct).

4. Every operator range in $F$ is either equal to $F$ or meager in $F$.

5. In particular, by Baire's theorem, if $F$ is a union of countably many operator ranges $U_n$ ($n \in \mathbb{N}$), then one of the spaces $U_n$ is equal to $F$.

My questions are :

1. Do all these facts remain true for ranges of operators acting on Fréchet spaces ?
2. What about countable increasing unions and decreasing intersections of operator ranges ?