はい、そうでした。 はい、そうです。 is 1 the only. 4+3=7 ですか? (is 4+3=7?) はい、そうです。 (yes, that is certain.) on the other. 山下先生のせんこうはれきしでしたか。 to answer the above sentence, would the sentences below be acceptable?
It's hot! hai is the distal form of yes,. Is はい the proper term to use to acknowledge. Disagreeing would get awfully complicated, but the answer should just be hai.
Atsui desu yo (はい。あついですよ), which means yes. I initially thought to reply with はい but that doesn't feel right, it feels like はい is more of a confirmation than simple acknowledgement. On the other hand, ええ is mild and polite but lacks sense of modesty and decisiveness compared. The two meanings mentioned in the question, はいどうず meaning here you are is #3, at least to my reading of it, and どうぞ meaning please as just a polite expression.
Despite how it might look on the surface, this is definitely not a question about tense. The key to whether to reply with はい or いいえ is not whether the form of the question is positive or negative, but whether the question is asked in a way which expects a. The difference is, はい is polite, formal and safe in every kind of conversations. そうです is a way of saying 'yes, that is certain.' for example, a discussion using the word might go like this: