Categories a computer scientist should know about I am a computer scientist with a CS degree from the 80's.
I am learning category theory by myself.
I am looking for advice about learning category theory.
E.g. it is quite helpful in learning category theory
to have examples from a familiar topic like logic or programming languages.
What I know
I have read the answers to this question and the paper
Physics, Topology, Logic and Computation: A Rosetta Stone
which builds up concepts with parallels between category theory,  logic, and theory of computation.
I know about the following categories:
• categories
• monoidal categories
• braided monoidal categories
• closed monoidal categories
• symmetric monoidal categories
• closed braided monoidal categories
• compact monoidal categories
• cartesian categories
• closed symmetric monoidal categories
• compact braided monoidal categories
• cartesian closed categories
• compact symmetric monoidal categories
Questions

Are there any other major categories that a computer scientist should know about?

Were there any that have fallen out of favor?
If yes, why?
Have they been replaced with other categories?

When thinking about categories should I
focus on the arrows first before the objects?
Note that most of the categories you considered are 'abstract', i.e., they require structure or properties of an abstract category. As computer scientists we should also be familiar several concrete categories which turn out to be useful:

Concrete categories


Set
Set$_{\mathcal P}$ (objects are sets, morphisms are relations)
Rel (objects are relations, morphisms are functions preserving these relations)
Pred (objects are predicates (= subsets) morphisms are functions preserving these relations.
Posets
$\omega$-CPOs
Nominal sets
Lawvere theories


Abstract categories


Functor categories
Presheaf categories, in particular, monoid actions, which are equivalent to finite automata iianm
Locally presentable categories
Monads
Enriched categories (in particular, O-categories)
Kleisli and Eilenberg-Moore categories
Embedding-projection categories
Profunctors


Also, sometimes properties don't commute on the nose, and the structure under consideration is of higher-dimensional (i.e., 2-categorical).

As to your question 3: you can define a category without mentioning objects at all (though it's conceptually clearer if we do mention the objects).