Why has the word συγγεής two γ? I know it comes from σύν + γεν, and that later the ν disappeared, but why putting two γ? And why has the ν disappeared at the certain point in history?

Because Greek didn’t have an ŋ letter, although they knew that the sound existed.

Phonetically, the final -n in prefixes was often assimilated phonetically to the following letter:

• syn ‘with’ + pathos ‘passion’ > sym-patheia ‘sympathy, compassion’
• syn ‘with’ + labē ‘taking’ > syl-labē ‘syllable: sounds “taken together”’
• syn ‘with’ + rhaphē ‘sewing’ > syr-raphē ‘sewing together’

Now if you put syn- in front of a velar, and the -n- undergoes assimilation to a velar just as it did to a bilabial or a liquid, then you would expect the n to go to an ŋ:

• syn ‘with’ + kopē ‘cutting’ > syŋ-kopē ‘syncope, cutting off’
• syn ‘with’ + grapheus ‘writer’ > syŋ-grapheus ‘author’
• syn ‘with’ + khysis ‘pouring’ > syŋ-khysis ‘confusion’

Those forms show up in Greek alright, but they’re written with a gamma where you’d expect the ŋ: <sygkopē>, <syggrapheus>, <sygkhysis>.

But we do have evidence that the gamma in that position stood for an ŋ after all.

1. In Latin, that first <g> was transliterated as an <n>: sygkopē = syncope.
2. There was no ŋ letter in Greek, so you would expect ŋ to be written down as a letter that sounded like ŋ—either <n> (same manner of articulation, not same place) or <g> (same place of articulation, not same manner).
3. The Greeks themselves said that that first <g> had a different sound, which they called agma; a fragment of Marcus Terentius Varro says that a grammarian called Ion had suggested agma should have been the 25th letter of the Greek alphabet. AGMA, A FORGOTTEN GREEK LETTER

ut Ion scribit, quinta uicesima est littera, quam uocant agma, cuius forma nulla est et uox communis est Graecis et Latinis, ut his uerbis: aggulus, aggens, agguilla, iggerunt. in eius modi Graeci et Accius noster bina G scribunt, alii N et G, quod in hoc ueritatem uidere facile non est. similiter agceps, agcora.

As Ion writes, there is a 25th letter, which is called ‘agma’, which has no shape, but a phonetic value that is the same in Greek and Latin, as in the following words: aggulus, aggens, agguilla, iggerunt. In words of this type, the Greeks and our Accius write a geminate GG, while others write NG, because it is difficult to recognize the real sound in the former; similarly agceps, agcora.