When searching for values, do I need to use all the columns specified in the partition key for a quick result?

Question

Let's say we have a chat room. Each chat room has several participants. In my case, it's one client and several operators (1 or more). Also, each chat room has its own specific theme. As we know in the Cassandra database, tables need to be designed based on queries.

Let's say I have such queries:

1. Get a list of all open chat rooms that don't have operators yet.
2. Get a list of all open chats that have a specific operator.
3. Get a list of all open chat rooms on a specific theme that don't have operators yet.
4. Get a list of all open chats on a specific theme that have a specific operator.

I know that denormalization is normal for Cassandra. But still, do I need to create separate tables for all these queries? I would also like to know how to check whether a value is included in the list in a SELECT query? As you can see, the operators column in my case has the list data type.

At this particular moment, for the first two queries, I created the following table:

CREATE TABLE IF NOT EXISTS "chat_rooms" (
    room_id uuid,
    created_at timestamp,
    updated_at timestamp,
    client varchar,
    operators frozen list<varchar>,
    last_message text,
    unread_message_number int,
    is_open boolean,
    theme varchar,
    PRIMARY KEY (
        (is_open, operators),
        updated_at
    )
) WITH CLUSTERING ORDER BY (updated_at desc);

For the third and fourth queries, I just changed the partition key and added the theme column. How correct is this?