Challenging questions, I like it.

Incoming wall of text!
I'll refer to the examples of police and fire fighters again to explain this point. Fairness in this context, is service rendered by the police and fire fighters without prejudice of any sort towards the citizens. Police and fire fighters would not be allowed to refuse to render help to someone based on their wealth, social status, ethnicity, or any other form of discrimination, that is what I meant by fairness without prejudice. Nix gave some good examples regarding how healthcare and social safety net in the UK work, which shows how more assistance and resources are given depending on how dire your situation is compared to others. I don't think I can explain it better then Nix's post, so I will leave it at that.
Right now in the field of medicine research, pharmaceutical companies are in direct competition with each other. As a result, their research and development labs most often do not share research data or analysis with one another until they have developed something that they can patent. The reason being, they do not want to give their competitor a chance to use their data and develop a new medicine for some illness or disease before they do, and get a lead on the market. Although progress are still being made, it is progressing at a slower rate than if there were more cooperation and information sharing. If they weren't in complete competition with one another, and if pharmaceutical research and development of drugs were socialized, with the sale of any development re-invested by society back into more R&D, all our best scientists would be able to work with each other and get more done.
Personally, I feel that society works best when there is a proper balance between capitalism and socialism, as going too far to the extreme on either side will result in inefficient progress. I'll give two hypothetical examples, one is a nation that is pure socialism with no free market, and another pure capitalism with no government infrastructure support.
Problem Example in pure socialism:
Lets say the government distribute out food in specific quantities to it's citizens, enough to match the daily recommended calories, vitamins, proteins, etc., so that everyone gets enough to stay healthy, but not over eat. But then some people have allergy to certain food, and would need to request a change in the type of food distributed to them. With a huge population, there will be too many food substitution request for the government to efficiently process, resulting in delays, and people end up going hungry for prolonged periods before their request are finally answered, while the food they are allergic to end up being spoiled and going to waste because it was still being delivered during that whole time and didn't get sent to someone else who could've benefited from it.
Problem example in pure capitalism:
A construction company specializes in building highways, and since the government isn't paying for the highway to be built in this pure capitalistic economy, the construction company make their profit by putting in road tolls so that anyone who need to use the highway have to pay, because after all, the roads are privately owned, not publicly owned. This isn't a major problem in of itself, but lets say Pepsi Cola decides to buy controlling shares in this construction company, and thus take ownership of all it's highways. Pepsi then decides that any vehicle transporting Coca Cola for the purpose of sale distribution isn't allowed to use their road, in an attempt to block off supplies of their competitor. Coca Cola responds by buying their own highway paving company, and proceed to buy up land and have parallel highways and inner city roads built so that they can transport their product. Aside from the obvious wastefulness of raw materials and redundant nature of having two sets of roads made, other problems occur for the economy, such as cost of houses sky rocketing because of less available land due to double road construction, and increase demand for construction labour which increase labour cost and construction cost.
These examples may seem extreme and silly, but then again, so is the idea of running society and economy based on a pure socialism or capitalism model.
As a final note, I want to further clarify all the terminologies, and their counter part.
Socialism is opposite to Capitalism/Privatization
Communism is opposite to Democracy/Republic
Hence, you can have a communist country that is using capitalism as its economic model. China is a good example as it has allowed for more privatization and moved towards more capitalism. Venezuela on the other hand is an example of a Democracy that leans towards socialism for their economic structure. The decision of what is the right balance between capitalism/socialism depends on the country and it's citizens, hence why we have elections to decide if we think our country is leaning too much to one side or the other. Although in the US, both main political parties are right-wing parties, one is just not as far right as the other.