What are capabilities of small Airplane in general?
You have a few questions and the answers to them vary greatly based on the plane in question. It's often said you should chose the plane for your "mission" which means you should think about what you would like to do with the plane and then chose the plane that best fits that "mission". But anyway let's address some of the questions at hand. 


  What is usual range of that kind of planes


This depends on the plane. Small planes can get anywhere from 500 nautical miles to upwards of 1500 nautical miles depending on fuel load and optional tank configuration. But the middle of that range seems to be the average. I have found that range is more dependent on how long you can sit in the plane not now much gas you can carry. Remember there is no bathroom on these little planes and some are not the most comfortable things in the world. 


  What is the worst weather under which it's ok to fly that kind of
  airplane?


Although this is changing most small aircraft are not certified to fly in ice. Some can be modified to do so and some come with the option when you buy it. That being said I don't know if I would fly a small plane in ice but IMC is possible and done all the time. Remember also that fancy equipment does not always mean you are safer, you are still flying a tiny plane in bad weather. 

Rain is possible but keep in mind rain on the ground could mean ice in the air depending on the temperature. Some small airplanes will be able to climb above this but that depends on plane. 

Wind is limited by the airports you are flying to and from. If the wind is blowing down the runway you are good to go but every small plane has a maximum demonstrated crosswind component that will limit your ability to fly in high winds at some single runway airports where a crosswind landing is the only option. 


  How expensive is to fly internationally with that kind of plane.


Again this changes from plane to plane and country to country. Small pipers can be flown for (from what I have read) under 10K(USD) a year and less if you don't fly very often. With older planes you will pay more in maintenance and less in up front cost while with a new plane you will pay more to buy it and less to maintain it as its fresh. You will also have to factor in gas and insurance. Insurance depends on YOU and your flying time so its hard to say what that will fall in. Gas here in the US (100LL) is around 4.56-6.00 where I am so figure in that based on fuel burn etc. 


  What is the procedure and usual cost for example if I fly from Rome to Prague, how the border control checks me, how and what kind of taxes related to that particular flight do I pay. Please, don't miss anything.


I don't fly in the EU so I will refrain from commenting here.


  If a plane is declared to have range of 1900 km, what is practical
  range in practice?


Here in the US you are required to flight plan with enough fuel for your flight plus 30 minutes of reserve. I don't like when companies put range because this is very much based on the winds and your course. If you get a 40 knot tail wind you will very quickly increase your range, while a 40 knot headwind will bring the number back equally as much. I would calculate range based on usable fuel and fuel burn of the engine. but if they are saying 1900KM I would plan for the 1750KM range to be safe. 


  If regular service for an airplane is declared to be after 7000 km of flight or X hours of flight, what has to be done usually in those services? Changing of oil, changing of filters?? How expensive usually is that?


Aircraft service is (as far as I know) done on hours not miles flown (although they are related in a way. Here in the US you need a yearly inspection which includes lots of things like testing gear and other components and ever XXXX hours when your engine hits TBO (to be overhauled) you need to overhaul it. This overhall can be costly but is very dependent on the engine. It can range anywhere from 15K to (insert large number here). You are also responsible to have anything fixed should it break obviously. 


  Cost to park 


There are 2 things here your storage cost and the cost when you actually go somewhere. I fly out of KPNE and have spoken to a few people about the cost of a tie down/hangar. It seems I can keep an airplane in a tie down for less than $100 a month but it would be outside all the time. Hangars get pricy and can be rented or bought. There are lots of small airfields where I am and I'm sure its possible to store a plane for cheaper than at KPNE. 

As for going somewhere that depends on ramp fees. Here in the US these fees are sometimes waived if you buy gas or very nominal in the first place. Some airports also have a small landing fee (KPNE is $5). I don't know about EU (although I'm sure they have it) but here in the US you can get the AFD (airport facilities directory) which has lots of info on these kinds of things. Most airports/FOB's now have websites that list their prices as well. 


  Anything else you think is important to know about possessing and airplane of that type.


It seems you are looking for a fixed number to a question that does not have an answer like that. You must first identify your mission then you can find a plane to fit the mission profile. You can then look at the prices of other people who have owned that plane and you may find what you seek but you will also find that there is a HUGE deviation in the prices as no two planes are alike.