Why does Lord Elrond call and refer to Mithrandir as Gandalf? Suggesting that it is because he is (only) half-elven would seem illogical, IMHO.
Not only Elrond uses 'Gandalf' rather than 'Mithrandir'. Galadriel too calls him 'Gandalf':
"Gandalf the Grey set out with the Company, but he did not pass the borders of this land." (LotR II 7 - The Mirror of Galadriel)
And Celeborn follows suit in the same scene:
"If it were possible, one would say that at the last Gandalf fell from wisdom into folly" (ibid)
Lothlórien's elves use 'Mithrandir' when singing in Sindarin:
Often they heard nearby Elvish voices singing, and knew that they were making songs of lamentation for his fall, for they caught his name among the sweet sad words that they could not understand. Mithrandir, Mithrandir sang the elves (ibid)
This evidence would support the idea that, like @Janus says in the comment, the Elves used 'Mithrandir' when speaking in Sindarin, and 'Gandalf' when using Westron.
Legolas addresses Gandalf as 'Mithrandir' twice, seemingly when he should have used 'Gandalf' (if the above hypothesis is correct), but both are exclamations of surprise, when one might slip into one's most accustomed mode rather than what might be considered "most appropriate". First, when they believe Gandalf dead and themselves followed by Saruman:
The old man was too quick for him. He sprang to his feet and leaped to the top of a large rock. There he stood, grown suddenly tall, towering above them. His hood and his grey rags were flung away. His white garments shone. He lifted his staff, and Gimli's axe leaped from his grasp and fell ringing on the ground. The sword of Aragorn, stiff in his motionless hand, blazed with a sudden fire. Legolas gave a great shout and shot an arrow high into the air: it vanished in a flash of flame.
'Mithrandir!' he cried. 'Mithrandir!'
'Well met, I say to you again, Legolas!' said the old man.
(LotR III 5 - The White Rider)
In the same conversation a bit later, when he is calmer, he uses 'Gandalf':
"It would ease my heart, Gandalf, to hear what befell you in Moria."
And again, when Gandalf comes to break the siege on Helm's Deep:
'Behold the White Rider!' cried Aragorn. 'Gandalf is come again!'
'Mithrandir, Mithrandir!' said Legolas. 'This is wizardry indeed!'
(LotR III 7 - Helm's Deep)
Gandalf himself says:
"Many are my names in many countries. Mithrandir among the Elves, Tharkûn to the Dwarves, Olórin I was in my youth in the West that is forgotten, in the South Incánus, in the North Gandalf; to the East I go not."
So, the Elves use the name that they know those around will know him by. If I have a friend called James who amongst my friends is known as Sheamus but I am talking to someone who I know knows him as James then I will say how is James or where is James. In the same way the Elves call him Mithrandir in their own language or when talking to people who know him as such, but will use Gandalf when talking to those who don't know him.
In relation to the second part of your question Gandalf is not Elvish at all, so being a half-elf is certainly not the reason. In fact Gandalf is a Maia, which is the equivalent of an Angel, served the Middle-earth version of Gods, the Valar, and was sent/came to Middle-earth to combat the power of darkness. Only the Elves know his true form.