I am new here, and I haven't got the time to read everything here yet, so perhaps part of the questions turns out to be unnecessary.
There are no stupid dumb questions, only dumb answers.
1. Is it known who built the Acceleration Gates? Evelopedia says nothing in particular, but I wonder how ancient it is.
The acceleration gates we use in-game are modern constructions. However, there are accounts in the PF of accel gates from the colonization era surviving into the end of the dark age - specifically in the Pator system, allowing the Minmatar to colonize several worlds. It's unlikely that any such ancient accel gates are active today. The ancient ones in Pator are probably scientific sites or museums... would make a brilliant in-game fiction marker, I think.
2. Stargates are relative novelties in the history of New Eden, built by the now known civilizations. Nonetheless I wonder how the initial colonists from the old Earth etc. transported themselves across star systems.
Well, EVE has three types of FTL travel: Gates, Warp and Jump Drives.
We know that the settlers built their own stargates, because A) they built the EVE gate to stabilize/replace the wormhole, and B) it's said that when the wormhole collapsed the stargate network was destroyed.
We know from accounts of the Amarr and Gallente space age that stargates were seeded by slowboating ships to new systems and having the crew on those ships wake up from cryostasis and construct the end-point gate. However, once warp and jump drives were invented I'm sure the pace quickened dramatically. At normal warp speeds, I think it's been calculated that an interceptor could go to a neighboring star in a week or so. Considering less efficient warp drives, a light ship might take a few months to get to a nearby star, then light up a cyno for the gate-builders to hop on over. You might even be able to do it with unmanned probes - just blast cynodrones in all directions and have the industrials jump in when they're ready.
The Terrans and Milky Way people were probably more advanced than the current EVE tech level when they started colonizing, so they would have had no problem seeding gates throughout the cluster.
3. In addition to 2, I am curious about a list of planets that were initially colonized by the first human settlers. Also, I like to know which of those colonies survived and which did not. Is this data known?
Well, no. We don't know all the planets that were colonized before the dark age. We do, however, know that the following civilizations arose separately, and we can pin original settlements to most of them:
The Jove (Probably in Utopia)
The Talocan (Possibly somewhere in Okkelen, although they were nomads so it's unclear if that's where they originated, or if they even had a homeworld)
The Yan-Jung (Deltole V and VI)
The Amarr (Athra/Amarr Prime, shared homeworld with the Takmahl, Khanid and Udorians)
The Gallente (Gallente Prime)
The Caldari (Caldari Prime)
The Ni-Kunni (Mishi IV)
The Ealur (Ealur system)
The Mannar (Mannar system)
The Jin-Mei (Lirsautton system, Chandeille and Chakaux)
The Achura (Saisio III)
The Intaki (Intaki Prime)
4. Here and there I read some stray notes on the development of fictive languages of the races in the game, but so far I did not really read about thorough linguistic endeavours (phonology, vocabulary, morphology, syntax, semantics) with a final result, let alone historical linguistic projects that link several stages of languages to the fictive history of colonies. Is this correct?
I've done some historical analysis of Amarrad, especially behind the scenes where I tie certain words and terms to different cultures on Athra (as a very rough guide, scientific terms come from Takmahl, civilized words come from Udorian, words about warfare and survival come from Khanid, etc). There's a link to the project thread in my signature. I welcome anyone who wants to chip in if they think they can contribute.
5. In addition to 4, I had the impression that so far there has not been written a descriptive grammar of Gallente/Gallentian, and that no reconstructed proto-languages are made. Is this correct?
I think this is true. The two biggest constructed languages are Napanii and Intaki. Amarrad is probably third, and there are plenty of smaller ones around (check the language sticky in this forum). I don't think anyone has touched Gallente. Possibly because it was so clearly stated that they were descended from French people that people have just used French as a substitute.
6. About questions 4 and 5: I am simply asking this befóre plunging into the deep with any ideas.
Happy to answer them. It's always wise to test the waters before jumping in. If you'd like to contribute to Amarrad, just post on the thread with your ideas, or PM me if you have questions about it.