It seems like 's from "Barbecue’s" is a shortened form of "is". Is there any rule for when to shorten it or not? And can I also shorten "was" as 's?

Native speaker’s answer
Rebecca
There are three cases in which you can use the "'s" contraction. For the first rule, the contraction must involve a subject pronoun, question word, or proper noun. It must be followed by a noun or present participle, so "was" cannot contracted as it is grammatically incorrect. Ex: He is = he's Ex: Barbeque is = barbecue's Ex: Who is = who's The second is the same as the above contraction but involves the word "has" instead of the present participle. Ex: He has = he's Ex: She has = she's Ex: What has = what's Lastly, the 's contraction can be used to indicate the possessive. Ex: The car of Lucy = Lucy's car Ex: The hat of my father = father's hat