You are very optimistic, Sergei!

In the countable case (or, only slightly more general: if there exists a cofinal countable subset) $\Lambda(P)$ is Fréchet, and you find many results about this case, e.g., in the book Introduction to Functional Analysis of Meise and Vogt, chapter 27. But even in this case, the characterization when $\Lambda(P)$ is reflexive or Montel (=Heine-Borel-Property) is a quite difficult theorem (this is called the Dieudonné-Gomes theorem). Of course, for Fréchet spaces barrelledness is for free, but I don't know of a characterization in terms of $P$ in the uncountable case (this is related to the explicit question at the end of your post -- my guess is that this is not always true: The hypothesis means that $\omega$ defines a linear functional on $\Lambda(P)$ and the conclusion means its continuity).

For the dual case of countable inductive limits of weighted Banach sequence spaces a lot of work has been done (e.g., by Bierstedt and others) to describe the dual again as weighted space and to characterize barrelledness in this situation. Again this is quite subtle, beyond the case of inductive limits of Banach spaces there are results of Vogt as well as Bierstedt and Bonet -- and if you really want to have a counterexample to your explicit question you should study their work.

Other than reflexivity or the Heine-Montel property there are many locally convex properties which are directly defined in terms of the semi-norms (Schwartz or nuclearity) -- for such conditions it is no difference whether $P$ is countable or not.

You are very optimistic, Sergei!

In the countable case (or, only slightly more general: if there exists a cofinal countable subset) $\Lambda(P)$ is Fréchet, and you find many results about this case, e.g., in the book Introduction to Functional Analysis of Meise and Vogt, chapter 27. But even in this case, the characterization when $\Lambda(P)$ is reflexive or Montel (=Heine-Borel-Property) is a quite difficult theorem (this is called the Dieudonné-Gomes theorem). Of course, for Fréchet spaces barrelledness is for free, but I don't know of a characterization in terms of $P$ in the uncountable case (this is related to the explicit question at the end of your post -- my guess is that this is not always true: The hypothesis means that $\omega$ defines a linear functional on $\Lambda(P)$ and the conclusion means its continuity).

For the dual case of countable inductive limits of weighted Banach sequence spaces a lot of work has been done (e.g., by Bierstedt and others) to describe the dual again as weighted space and to characterize barrelledness in this situation. Again this is quite subtle, beyond the case of inductive limits of Banach spaces there are results of Vogt as well as Bierstedt and Bonet -- and if you really want to have a counterexample to your explicit question you should study their work.

Other than reflexivity or the Heine-Borel property there are many locally convex properties which are directly defined in terms of the semi-norms (Schwartz or nuclearity) -- for such conditions it is no difference whether $P$ is countable or not.