I'm glad you asked. It's so easy for a person to get unnecessarily offended by language subtleties in a non-native language. I remember once trying to say to my mother-in-law in my pidgin German, "I'm ready to visit your cousins" and she understood "I'm fed up with your cousins." It took us a long time to straighten that out! It turned out there was an idiom that I wasn't aware of, and she forgot that idioms were waay beyond me.
I will give you my person reactions, as a native English speaker from the U.S., to the various expressions.
(1) Care to add some references for your claims?
Reaction: the way this is posed doesn't sound rude to me. However, the person who wrote this is standing up to you, saying "Hey, friend, put your money where your mouth is, and tell us how you arrived at this!"
(2) Please, care to add some references for your claims?
This doesn't sound like a native English speaker. (1) was informal, and the informality softened it. By putting the formal "Please," at the beginning, it becomes confusing, like someone asking you if you'd like some hot sauce with your chocolate cake. They just don't go together.
(3) Would you care to add some references for your claims?
The tone of (3) is so much more formal than (1) that it gives me the feeling that the writer is feeling either offended, or confrontational, or both. Sometimes an abrupt increase in formality signals discomfort on the part of the speaker.
Bottom line, it is easy to take things badly when you don't have the benefit of knowing the person already, seeing their body language, or hearing their intonation. And language differences only complicate things more. Just keep all that in mind, when you participate in these international forums where English is forced down everyone's throat, so you can keep things in perspective.