Is a covered reshut ha-rabbim still considered reshut ha-rabbim?

Does a public domain (reshut harabim) or private path separating two houses prevent combination for zimun?

Zimmun — Joining and Splitting Groups

Is reciting Shehecheyanu upon acquiring a new house or vessels obligatory or merely optional (reshut)?

Blessing on Seasonal Fruits

Is Shehecheyanu a mandatory (chova) or optional (reshut) blessing on new seasonal fruit?

Shehecheyanu

If a Jew rents oxen to a non-Jew to plow with them, and the non-Jew does plow on Shabbat — what determines whether the animals are still considered 'the Jew's' for purposes of shiviyas behemah?

Shabbat Candle Materials

When a pot on a kirah with coals is covered with cloths that touch the pot, is this considered hatmanah even though the cloths themselves do not add heat?

Insulating Food (Hatmanah)

Where may one carry stones for wiping on Shabbat — only within a private domain (reshut hayachid) or also from a karmelit?

Handling Stones and Raw Materials