Translation:We do not exist.
I believe Czech never uses double vowels (for example, ee) to indicate a single sound. Some languages use them to show a long vowel, but Czech would write é instead.
So I think that two vowels written together in Czech must always be separated by a glottal stop (short break).
you are right concerning identical vowels, but of cause there are combinations of different vowels that are not separated by a glottal stop. e.g. the eu in "museum", whereas "nealcoholický" has a glottal stop between the ne- and the rest of the word. The decisive criterion is the grammar: if the word is combined of different parts that come together exactly between the two vowels, then there is a glottal stop.
Czech recognizes only three diphthongs: /ou/, /au/, and /eu/ - and /ou/ is the only one that appears in domestic words. Thus "touha", "auto", "euro" are all pronounced as two syllables, with no glottal stop.
Whenever any other two vowels appear next to each other (like in "neexistovat"), they are separated by a glottal stop (ne'existovat) in careful speech. But in "lazy" speech, this glottal stop is often omitted. In practice, it doesn't matter how you pronounce two consecutive vowels - it never changes meaning whether you make that "short stop" between them or not - and different native speakers have different habits in this, often depending on how clearly they want to speak.
Slovak also writes "ne-" (pronounced /ňe/) together with verbs: "nemám", "nechcem", "nemôžem", etc., except for the verb "to be", which for some reason has a different negation: "nie som", "nie je", etc.
The situation is nearly inverted in Serbo-croatian and Slovenian - they write "ne" separataly: "не радим", except for the verbs "to be" and "to have", which are written together, because the resulting combination changes: "сам" -> "нисам", or "имам" -> "немам", etc.
Polish, Russian and other Slavic languages write "ne"/"nie" separately everywhere.
The Baltic languages - Lithuanian and Latvian - also use "ne-" for negating verbs and they write it together with verbs, like Czech: "neturiu", "nedirbu", etc.
Czech and Slovak as well as Latvian all have fixed initial stress, and since this stress shifts onto the "ne-" in negated verbs, it makes sense to write it together, because prosodically it becomes one word. This, however, makes no sense in case of "nie som", "nie si" etc. in Slovak. And Lithuanian has free stress, so I can't explain it there. This is very different in Serbo-croatian, where the stress does not shift onto the "ne", so it makes sense to be written separetely: "(ja) RAdim" (I work) -> "(ja) ne RAdim" (I don't work) - compare with Czech "PRAcuju" -> "NEpracuju".
Also, as an analogy, Czech also uses the "ne-" prefix to negate nouns and adjectives, e.g. "nezdravý" (unhealthy), "nekuřák" (non-smoker). It's the same prefix and it's written together with nouns and adjectives, just like it's done with verbs.