I think I will contribute with a listing of existent animals:

• Those that belong to the emperor
• Embalmed ones
• Those that are trained
• Suckling pigs
• Mermaids (or Sirens)
• Fabulous ones
• Stray dogs
• Those that are included in this classification
• Those that tremble as if they were mad
• Innumerable ones
• Those drawn with a very fine camel hair brush
• Et cetera
• Those that have just broken the flower vase
• Those that, at a distance, resemble flies

That's stolen from somewhere, I forget where.

Borges.

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I guess two confuses you, and three - don't forget four and the rest of them. And love justice and The American Way. Superman, unicorns, dragons, all of the English and French kings - they do not exist, do they. These have no existence? Maybe we should pause here: answer: do these exist, yes or no? — tim wood

No they do not exist. In the common usage of the word "exist", which I am familiar with, fictional things do not exist. Nor do love, justice, and other ideas exist. They are conceptual only, and therefore not existing things. Notice that love and justice refer to relations between things, while concepts and imaginary fictions are ways of thinking. None of these are themselves, things, and that's why we cannot class them as existents.

Is the problem "encounterable"? Let's consider that no one "encounters" anything at all, except mediately through perception and idea. And by that standard, unicorns and their like are more purely existent than any of the furniture of the "real" world, being pure idea undiluted by perception. You really are not making sense. Why is that? — tim wood

It's you who's really not making any sense here. I simply rejected "encounterability" as the defining feature of existent, for very good reason which I explained. I don't know what you're trying to say.

If the existence that is necessary is particular to an instance of time is also saying that there are other instances when it's not, then it would seem to be contingent on the time — tim wood

Short version.....
That which exists being contingent on time implies everything which exists is contingent on time. If everything is contingent on something, we say that something is the condition for all those things contingent on it. It is accordingly we say time is the condition of all that exists. All this does is relieve us of the need of a quantity of time for the existence of things in general, while requiring a certain time for things in their relations to other things.

Long version.....
Trouble is, the condition for a thing doesn’t tell us what we want to know, which is what the thing is. If the time is the condition that makes everything possible, all we need are the conditions that make everything describable. The only way for us to describe things is by means of the concepts that we can logically apply to them, and because we are not describing time, we don’t need to think of time as a concept.

But the things we wish to know about must first be determined as describable, in order to be certain any of our concepts can ever be applicable to them, which effectively grants the possibility of knowing what they are. It would be a major evolutionary disadvantage for us to have a describing system that cannot tell itself the thing attempted to be described never was describable in the first place. Enter the categories, those pure concepts arising spontaneously from the system itself, which serve as the criteria against which all the things we wish to know about, become describable. We find, in order to be described, a thing must exist, so existence is a category; a thing must be real, so quality is a category; a thing must consist of something, so quantity is a category, and finally, a thing must be either a cause or an effect, so relation is a category. To name four of the twelve.

From here it is a short hop to understanding why we don’t need the categories for what we think, because the thought is the description and is infallible, and why time is not a category because it doesn’t set the ground for describing by means of concepts. In addition, time is divisible but the categories are not, insofar as different quantities can be attributed to time, but i.e., necessity, cannot be quantified at all, which is sufficient in itself for claiming conditions for everything and the conditions for describing everything must be irreducibly distinct from each other.
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Would you agree (with me) that this is grounded in contingency? Or at least there's some work to be done to either refine or qualify "necessary existence"? — tim wood

Absolutely. There is nothing whatsoever that isn’t contingent, because the totality of our knowledge for everything is impossible. But, like I say, the human system is inherently circular, as what I just said, proves. I contradicted myself by stating a logical truth. That everything is contingent is necessarily true is self-contradictory.

(Oooooo....transcendental illusion!!! Now there’s a rabbit hole for ya!! (Grin))

Hey....we do the best we can, right?

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Of necessarily and ordinarily existing things several question arise. (...) Are they both sub-species of existing things? Is one included in the other? Or is necessary existence a separate genus? — tim wood

I don’t know how we’d be able to tell the difference between an ordinarily existing thing and a necessarily existing thing. But then, we don’t say...that which exists, exists ordinarily. So maybe there is a difference, or, existing ordinarily doesn’t make any sense to begin with. Dunno.
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In different words, if existing, then existing necessarily. — tim wood

That’s the entire logical argument in a nutshell. It makes no difference what the things are, but only if there are any, and how it is that the logical argument is true.
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That is, is the world altogether accessible to reason? I'm obliged to think it must be. — tim wood

Science says it is. Or it used to, until it was proved there are things we are just not equipped to know. I’ll go with....of the sum of reality empirically accessible to us, it is equally accessible to our reason. I’m a YankeeVirgoBabyboomer, so not known for my optimism.
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Addendum:

I'm looking to ordinary language for guidance, and it strikes me that whatever the reality is, either it yield to language, or language to it, and with respect to reality and the real, the reality is prior. — tim wood

Reality is prior, and language yields to it, or, which is the same thing, language yields to what we think reality is. The vast majority of human thought is by means of image, as should be quite obvious, language having no occassion but for subjective introspection and objective communication.

I guess you did not read the OP. — tim wood

I've read the op, but as I've said, I don't agree with your principles of categorization. I think your expressed principles display a lack of understanding of what it means to exist. That's why I suggested that we ought to clear up the issue of what it means to exist before we attempt any such categorization.

But this is all useless. Clearly you are unable to take part in the general point of the thread, and you have nothing to offer but that "which I am familiar with." — tim wood

How do you suppose that I could offer you something other than what I am familiar with?

I haven't yet been able to find "the general point of the thread". It appears like your intent is to define "exist" in some odd sort of way, and I can't understand why. Despite the fact that I've made some suggestions to help you to express yourself, you haven't been able to clarify at all, what you're trying to do. So I remain lost and confused. Now your argument has digressed to ad hominem premises drawn from my confused state. If you are completely disinterested in what I am familiar with, and your intent appears misleading to me, how can we find common ground?

I'm looking to ordinary language for guidance... — tim wood

No, you have clearly expressed that you are not looking to ordinary language for guidance. Your intent appears to be to offer guidance through an abnormal use of language, and this I class as misleading.