In the examples you give, your phrases are correct.
In your examples, “there is” is correct because it is followed by a series of singular nouns “there is a sandwich, a potato and an orange on the table”. “There are cats and dogs” is correct because the nouns are both plural.
Your example concerning a runny nose is also correct. “Runny” is an adjective and “running” is a verb form. “Running” is sometimes used as an adjective, as in “hot and cold running water”. However, at your students’ level I don’t think it’s necessary for them to know this distinction.
Unfortunately, I think the Junior high school teachers are not aware of or are not teaching the correct usage. This is a very tricky situation because you have taught the correct forms, the students have learnt them and the teachers are giving grades to something that is wrong. If you point this out to your students, it could undermine the authority of their high school teachers. If you go along with what these teachers are saying, you’ll be teaching your students something that is wrong and changing what you previously taught. It’s a delicate situation that you’ll need to handle diplomatically.