You are right that the past of both 'I have' and 'I've got' is simply 'I had'. The form 'I have got' - or more usually 'I've got - is an informal alternative to 'I have' but only in the present simple tense. In all other tenses, we just use the verb 'have' without 'got'. That said, the sentence 'She said that she had got a lot of homework' is correct. ( I'm presuming you mean 'homework', by the way - there's no such thing as 'hometast'!) The meaning is slightly different, however. 'I had got' is a past perfect construction (not a past simple), and the main verb here is 'get' rather than 'have'. The meaning, in this case, is 'I had received'. So, if you say 'She said she had got a lot of homework', it would be the British English equivalent of the American form 'She said she had gotten a lot of homework'. We'd understand this to mean that her teacher had given her a lot of homework. If this were an exam, I would expect the correct answer to be 'had', but I would also accept 'had got', as this has virtually (though not exactly) the same meaning. The meaning is practically the same, but the tense is different. In American English, 'had got' would probably not be acceptable here.

"I have got" simply means "I have". I'm not American and I'm not a native speaker, but I've heard lots of Americans use "I've got" or some similar structure, as in "I've got a lot of homework" or "you've got to be kidding me!", or in even more familiar turns of phrase like "I gotta go" (= "I've gotta go" = "I have got to go" = "I have to go").

Correct grammar is 'I have a lot of homework to do'. The word 'got' is extra and unnecessary here. We say it a lot but, when I was at school, I was not allowed to write it. (UK)

It's OK to say "had got". It is possibly not used as commonly in US English but we would need a US English speaker to confirm this.