Dharma Name in tibetan tradition I took refugee and got a dharma name.

Why do I get a dharma name?
How does a Lama usually choose the name?
Do I make use of this name in Dharma related context? (Like for example writing a letter to a Lama or maybe even introduce myself)
Best and thank you for answering.
Like you, when I took refuge, I received a dharma name—and so did the other twenty, or so, people who took refuge at the same time. We all received the same dharma name. I can’t answer your questions 1 and 2, but I do have a first-person account of a teaching related to dharma names in Tibetan Buddhism.
I once attended a weekend course on meditation in New York City given by a well-known Tibetan lama, and at one point, he digressed to tell a story about what are called “Buddhist names” and a time that he tried to teach a student that his desire for such a name was an attachment that was holding him back.
He related that this student had come to him to ask the lama to bestow a Buddhist name on him, pointing out that he had been a good and earnest student for many years. The lama asked the student what his given name was, and the man replied, “John.” The lama said: “Ok, your Buddhist name is John.” The student, as the lama recounted, stood there bewildered. So the lama, after a sufficient pause, explained to the man that his idea of a Buddhist name was an attachment.
I found the story the lama related very useful myself, especially given the predilection of many people to become attached to symbols and traditions, rather than the underlying wisdom that they stand for. I don’t know what the purpose of a Buddhist name is, but I believe it may be more a way to fulfill a ‘Western’ need, than a substantive accomplishment.
It’s different, of course, if you have a more direct relation with your teacher, and he gives you a Buddhist name. In that case, it is a direct recognition of your efforts along your path.