stay at 不是动词短语,只是常见的动词+介词搭配 stay in是动词短语,意思为逗留。如: I will stay in Beijing for a week. 我将在北京呆(逗留)一周。 Don't stay at the roof. 别在房顶呆着。
昨天你呆在家里吗? Did you stay home yesterday? or Did you stay at house yesterday? 上面两句中home为副词,house为名词。home也可以做名词用,但意义上等于house+family。 “昨天你呆在家里吗?”这个语境只是在问你昨天有没有在“家”这个地方,所以用该用house。 如果你要表达“和家人呆在一起”可以说“stay (home) with my family/families"