Why the "есть" here? What does it add or emphasize?
есть means is.
Yes, but we usually omit есть in a sentence like this.
If i understand it correctly "есть" is used to indicate the existence of something. Since we didn't know the park existed before, "есть" is necessary.
So from what I understand: without есть would be "The park is near the house." With it, it's "There's a park near the house."
Near the house is a park is not accepted.
I would report it.
Does it roughly translate to "in the vicinity/neighborhood of the house there is a park"?
Yes. That is an acceptable, literal translation. DL should accept it but often does not accept literal translations, for reasons I've never figured out.
Can "есть" be omitted here?
Yes. See comment above.
'There is a park by the house' is marked incorrect but I don't understand why.
"Next to" is not accepted. Is this a case when only "near" is acceptable?
This is an unfair one, you stated "рядом с" = "next to"...