If the parents of a ger pass away, must the convert sit shiva for them? If so, is it preferable to sit shiva in the mourner's hometown or the parent's hometown?
See here from Gershon Gold:
The Rambam in Hilchos Avel 2:3 says that a Ger is not obligated to mourn for either of his parents. This is so because someone who is a Ger is considered as if he is reborn, and therefore has no Halachic relationship to his parents (Yevamos 22a; Bava Kamma 88a). The Beis Yosef (Yoreh De'ah 374) quotes the Mordechai in the name of the Ri that a convert must mourn for his mother, but the Rema in Shulchan Aruch Yoreh De'ah 374:5 explicitly disagrees with this view.
My Rosh Yeshiva Rav Gedalia Anemer zt'l from Silver Spring Maryland , held in accordance with the Rema and ruled that a convert should not sit shiva for their parent.
Rav Ovadia Yosef (Yechaveh Da'as Vol 6 Responsa 60) however rules that a convert may sit shiva for a parent.
It is common for those sitting shiva to do so where they address comfortable. Many travel to different places to allow for family or friends to visit. However call your local Orthodox Rabbi to clarify if this is the proper thing to do.
Even though a convert has to honour his biological parents, he doesn't mourn for them.
More than that: If an entire family converts together, they do not mourn for each other.
Sources from the Kitzur Shulchan Aruch:
סעיף כב' גֵר אָסור לְקַלֵּל אָבִיו הַגוֹי וְלֹא יְבַזֵּהוּ, שֶׁלֹּא יֹאמְרוּ, בָּאַנוּ מִקְּדֻשָּׁה חֲמוּרָה לִקְדֻשָּׁה קַלָּה, אֶלָּא נוֹהֵג בָּהֶם מִקְצָת כָּבוֹד.
סעיף ה' גֵּר אוֹ גִּיֹּרֶת שֶׁנִּתְגַּיְרוּ עִם בְּנֵיהֶם, אֵין מִתְאַבְּלִין זֶה עַל זֶה, דְּגֵר שֶׁנִּתְגַיֵר, כְּקָטָן שֶׁנּוֹלַד דָּמֵּי, וְקֻרְבָה שֶׁהָיְתָה בְגוֹיּוּתָם אֵינָהּ קֻרְבָה.