1. ... All my potential locations were suitable for, let's say, resarch used observatories - they were on higher altitudes with clear horizonts, but were far from cities so nobody but astronomers would ever come to visit it. ...
My questions would be - how big should the telescope be?
My suggestion would be about a half meter ... Something like an RCOS or PlaneWave 24-inch scope is large enough to provide a Wow factor at the eyepiece yet still fit in a 20-foot diameter dome. Both are readily available commercially. The price tag for this combination will be in the $250,000 (USD) range, including quality construction. On the other hand, a 16-inch Schmidt Cassegrain from Meade, or a 14-inch Cassegrain from Celestron, including a smaller dome and quality construction, would be well under $100,000 if that's a more suitable budget.
What are the size ranges of amateur and professional telescopes? (OK, I know that 3m telescope is not amateur :) ).
The 200mm SCT is the most popular size of commercially available scope for amateurs. But that size is not large enough to give much of an impression when observing something like galaxies and nebulae. Your starting point should be about twice that. For typical public outreach programs in the US, many clubs have public viewing nights in public parks. Their members bring their telescopes and share with the public. It's pretty common to find at least one or two 300mm scopes at such events, and it is becoming more common to see something like a 400mm dobsonian scope there.
Typical dobsonian scopes are not well-suited to observatory placement. They're made for easy assembly/disassembly and are quite portable even in sizes up to 400mm to 500mm. In sizes above about 300mm, the average person needs a stepladder to reach the eyepiece ... hence these are not the best design for access by a broad range of people.
An SCT has its eyepiece on the bottom end, so it can be made accessible to those who must remain seated for observing. The periscope-like device that is used for accessibility by wheelchair-bound people works quite well with this design.
What are the benefits of bigger telescope? (e.g. Do you see further or clearer than with a smaller one?)?
A telescope gets it[is to gather light. The larger its aperture (diameter), the more light it gathers. The more light it gathers, the better it sees dimmer objects. Most galaxies and nebulae are too dim to see well in small scopes. The brighter planets (and the Moon, of course) do better with more magnification. Therefore, the SCT design (which packs a lot of focal length and aperture into a physically small package) is quite popular.
How big should the dome be?
For public access, as large as possible. Ash Dome models are available up to about 28 feet in diameter ... almost 10 meters. Such a dome can accommodate about 25 to 30 people around a large amateur-class SCT on its pier without overcrowding. At Comanche Springs Astronomy Campus, where I work, we have such a dome, equipped with a 15-inch (~400mm) refractor. This is not the best choice for a public facility, but it was available to us at a good price. Such a telescope is quite impressive to look AT (if not as impressive as a larget SCT to look THROUGH), and the public really enjoys it. You can see it and the observatory at our Web site: www dot 3RF dot org), or click here.
I would suggest a minimum of 12 feet diameter for a dome for public use. You can shuttle people into the dome in groups of about a half dozen pretty readily.
Can the dome floor be on the same level as other rooms (e.g. control room, maybe classroom, laboratory, office... etc.). I'm asking this because I've read that observatory walls should have low thermal mass which creates some issues about integrating it with other parts of the building.
For public use, yes. This is why the Griffith facility has domes at the ends of the building. Low thermal mass is more important for research purposes. Public observatories often include air conditioning and heating, set appropriately for the season for observing rather than for comfort: enough hit in the wintertime to offset frost, for example. But for best observing, you want the inside air to be at the same temperature as the outside air. There is an effect called "dome seeing" (air turbulence induced by body heat and building heat) you want to avoid. The easiest way to do this is to minimize the number of people in the observatory (under the telescope) at one time, and to use a large dome shutter opening (to release heat to the atmosphere more readily). Most observatories I've seen use exhaust fans or very large doors at the four compass points, to move air through the dome more effectively. Generally, the idea is to keep the telescope objective or mirror (which can have very large thermal mass) as near as possible to ambient outside temperature, and to prevent rapid and large swings in temperature.
3. I'm aware of this, and I have acquired general EU regulations for people with disabilities. I'll aim to satisfy all of the requirements, and I've actually found manufacturer that provides instrument that allows handicapped access to the eyepiece for most telescopes.
I've seen the "inverted periscope" designs, which work best with SCT type scopes. To make large telescopes wheelchair-accessible, some facilities use ramps or lifts. It will be important to use an angled-fork mount to allow the eyepiece to be offset as far as possible from the pier ... for wheelchair accessibility.
For the end, the most stupid question of all :) What do the astronomers do during the day in observatory, or when they can't observe the sky due to bad weather?
Most observatories have a good library(!). For daytime use, consider a solar telescope. These work best when the observatory itself is closed to the outside air. In one popular design, a mirror called a heliostat outdoors reflects an image of the sun through a glass port or a tunnel into a darkened room, through a proper solar observing filter (these are either a white-light filter or an etalon or other hydrogen-alpha filter) indoors for observing. For a white-light telescope, the image can be projected onto a wall or a tabletop for direct viewing. The Mount Wilson and Kitt Peak public solar observatories use this method. But h-alpha filters are quite dark and generally you view the image from a digital camera, on a computer monitor.
The h-alpha design is more difficult to build as an indoor/outdoor setup, so most of these that I have seen are set up totally outdoors, with a portable shade used to minimize intrusion of stray light.
And, of course, the planetarium shows are your biggest draw for cloudy days/nights, or during the daytime.