Why did he address Eliezer as 'Baruch Hashem' - bearing in mind that he was a Cana'ani and therefore cursed?
Targum Yonasan: He actually mistook him for Avraham. 1
He ceased to be cursed, and became blessed, because he served a Tzadik (Pesikta Zutresa), or because he began with Baruch (Baruch Hashem - Midrash Agadah).
Hadar Zekenim: It is because he served Avraham faithfully.
And that would also explain why he told him that he had cleared the house of Avodah-Zarah, as Rashi explains.
What made him decide to invite Eliezer at this point?
Seforno: When he saw the ornaments that Eliezer had given her, he realized that not to invite him would constitute a lack of gratitude, plus the fact that he had asked to stay with them, as Rivka had informed them. 1
In fact, he made that decision already before running out to him, which explains why he told him that he had cleared the house to accommodate him and his men, as the Seforno himself explains.
What are the implications of "Anochi Pinisi ha'Bayis"?
Rashi: It implies that he cleared the house of Avodah Zarah. 1
See Torah Temimah, note 24, who explains how this is implied
If Rivka ran home to tell her mother about Eliezer, why did Lavan invite him in?
Oznayim la'Torah: This teaches us the difference between the two siblings; Whereas Rivka would not drean of inviting a guest into the house without consulting her parents, Lavan had no compunctions about doing so. 1
Oznayim la'Torah: Just as he had no compunctions about answering before his father
רש"י: פניתי הבית- מעבודה זרה: למה לא פירש כפשוטו?
גור אריה: רבקה כבר אמרה שיש אצלם מקום ללון, ולא הוצרך לפנות לשם כך את הבית.