Do general truths entail existence?

You are basically restating an hundred-year old problem of logical semantics that has occupied the time of famous philosophers like Bertrand Russell. His example sentence was "The kind of France is bald." Various solutions were proposed within logic including the 'possible worlds' theory and the 'sense/reference' distinction. @user6726 is right, you would get a far better answer to the current thinking on this on Philosophy SE.

However, this is also a linguistics problem. At least from two perspectives.

1. Linguistics shares referential theory of meaning and its problems with philosophy. Many semanticians have held some form of the truth-conditional theory of meaning. I.e. the meaning of a statement is its truth value. This is incredibly useful in formal semantics and computational approaches to language but (in much current thinking) a complete dead end for thinking about meaning. So, by bringing up a question like this, you are raising a disciplinary debate about the nature of meaning in language.

2. Purely and uncontroversially within linguistics, your examples raise questions of speech acts and presupposition which are two concepts introduced into thinking about language at least in part as an antidote to the truth-conditional view of semantics. The core of the idea is to look at the communicative action performed by a statement rather than just whether it is true or not. In this way, the truth conditions are irrelevant to most language that is spoken. Speakers' intentions, context, assumed knowledge, etc. - all of those are more important to understanding an utterance or text than whether we can map it onto the world or not.

Many of your examples were addressed directly and very powerfully by Michel Fauconnier in Mental Spaces which he later developed with Mark Turner into the conceptual integration or blending theory. I highly recommend their The Way We Think. George Lakoff has enriched mental spaces with his notion of idealized cognitive models or later frames which also address many of your questions. I recommend the first few chapters of Women, Fire and Dangerous Things to understand the depth of the challenge to the truth-conditional approach to meaning which your question assumes. A more complete elaboration of these issues can be found in Leonard Talmy's cognitive semantics. Look in Part 2 of Volume 2 at semantic interaction for treatment of the issues you raise.

With a richer conception of meaning than referential truth, you will find that your problems simply disappear but in that will open a far more complex and interesting world of meaning making in language than Frege, Russell and the like could grasp with their riddles.