He told me to come to his house and do the remaining assignment. ok. You just had the wrong preposition.
He told me that I should come to his house and do the remaining assignment ok..You should only use quotation marks when it is the exact words spoken. He wasn't telling you that he should go to someone else's house...
The 1st sentence should be...He asked me to come to his house and do the remaining assignment...
The 2nd Sentence should be....He suggested me that I should come to his house and do the remaining assignment... Even If you don't use "me", it's fine...