[Fordim Hedgethistle]

You will need to give me a well thought out explanation of why the Ancient Elf/Host’s brother had decided to refuse the call of Mandos after his death at the hands of the balrog and now wanders as a Houseless fëa. What makes him so different from his brother - who I’m sure would never refuse Mandos’ call should his body die?

As it happens, I've been wondering why any Elf would refuse the call. In the case of this fellow, however, I would think that he refused it from a mixture of pride and the refusal to accept change (which were the downfall of all-too-many Elves in the First Age). First, he died when the threat from Morgoth to Middle-Earth was still very real and he was still very attached to it. Being of the Firstborn but not one of the Noldor, he had already made his decision in favour of Middle-Earth rather than the Valar a long time before. He decides to remain in Middle-Earth from a refusal to accept that his time to leave the story had come; and he was proud enough to believe that he could resist the powers of darkness. He even thought that he could lend the Elves some aid in their wars, and for a time he was even able to do this to a certain extent by hindering the wills of their enemies and 'whispering' (so that it felt like inspiration or instinct) to his friends.

And why would the presence of this Houseless Brother in the Ancient Elf’s body allow them to travel with the orcs to Dol Guldur?

Whose influence is the Houseless Brother's fëa under? Sauron?

The Houseless is under the control of Sauron, but his pride is such that he thinks that he is fooling the Dark Lord, and he is able for a time to convince his brother-host that they are merely 'playing along' with Sauron's plans. In reality, Sauron has decieved the Houseless into seizing control of his brother in order to bring back someone to Dol Guldur who will be able to provide the Easterlings with information about the Lorien defenses. The orcs and Easterlings in the raiding party are very much aware of this plan and of their mission to return the Elf to Dol Guldur unharmed (mostly).

And what do you have in mind for Galadriel’s intervention?

I don't imagine that Galadriel will be able to do anything -- the host Elf, though, is hoping that she might. I am assuming that with the fall of Sauron his control over the Houseless, like his control over all of his slaves, will be broken and the Houseless will be 'free' once more.