2 doubts in db design
I'm not familiar with the layout of your database design (there are many layouts in the market). But what I'm figuring out is that you are trying to connect two entities together, directly. Which is wrong. Two entities can only be connected by a relationship (no matter what ER layout you are using).

Your first query can be subdivided into three parts:
1) An event can have only one organizer
2) Organizer is an instance of User
3) There is a one to many relationship between event and organizer.

So you have three entities: Event, Organizer, User.
Then, if you have studied about weak entities, you would want to convince yourself that organizer is a weak entity which is dependent on the entity User. Connect entities User, and Organizer with a relationship "is_instance". Now you are done with the entity User. 

Let's now move on to the relationship between entities Organizer, and Event. You already figured out that it has a one to many relationship from Event to Organizer. So great!!

Now coming to your next query: An event can have at most three categories but a category can have any number of events. You can say that this is a kind of many to many relationship. Some ERD allow you to explicitly specify the cardinality between the entity and the relationship (as far as I remember it is done by specifying the number over the arrow joining the entity and relationship). So you can model this as joining the entities Event, and Category by a relationship has_category and specify the range 1..3 over the line joining Event and has_category.

Extra clarification:
There is always a many to one relationship from a weak entity towards its underlying entity. Your inclusion of details to the entity organizer and the dependence of Organizer's ID on User's ID makes it a separate entity.

(Note: due to lack of information, I assumed that there may be more than one value for the attribute "detail" for the same user in the entity Organizer. If this is not so and that there is only one tuple in the entity Organizer for every ID such that every tuple in the entity Organizer is unique on attribute Organizer_id, then you could also choose to make the Organizer entity as a strong entity and have a total participation of entity Organizer in the relationship between User and Organizer)

Additionally, your query also states that the user can participate in many events, so there can be a many to many relationship between entities User, and Events.

Hope this helps.