Are Lusin and Souslin spaces sequential or even Fréchet-Urysohn?

Question

First some definitions:

• A Polish space is a separable and completely metrizable topological space.
• A Hausdorff space is Lusin if it is the image of a Polish space under a bijective continuous map.
• A Hausdorff space is Souslin if it is the image of a Polish space under a continuous map.
• A topological space is sequential if every sequentially closed set is closed.
• A sequential space is Fréchet-Urysohn if the sequential closure of every subset is sequentially closed (and thus closed).

Wikipedia says that "A Lusin space is a topological space such that some weaker topology makes it into a Polish space."

So I think this is wrong and should be restated as: "A Lusin space is a HAUSDORFF space such that some STRONGER topology makes it into a Polish space."

Similar for Souslin spaces in the Wikipedia article where the Hausdorff property for Souslin spaces should be added.

Now for the main questions:

1. Is every Lusin space Fréchet-Urysohn or at least sequential?
2. What about the larger set of Souslin spaces? Are these Fréchet-Urysohn or at least sequential?

If 1. or 2. is wrong what about the special type of such spaces that arise in functional analysis as locally convex spaces that are countable locally convex inductive or projective limits of Fréchet or Banach spaces and their weak* duals?