There is/are progress and improvements.

is progress, use the word closer to the verb, but why not just cut the "s" off of improvments in the 2nd sentense?

I agree that IS sounds better, and removing the 's' would help; but what if the sentence were turned around:
Progress and improvement IS or ARE...?????

it's there are progress and improvements because improvements is plurial therefore are is used when there are pluriels.

Progress is singular and Improvements is plural so one would have to say:

"There is progress and improvement." or "There is progress and there are improvements."

When you have a compound subject connected with 'and', always use the plural form of the verb.

When you have a compound subject connected with 'or', match the verb to the closest noun.

The sentence "There are progress and improvements" is in passive voice; the "correct" way to say it would be (although few would use it in the real world), "Progress and improvements are there." Or, "Improvements and progress are there."

Just kick out the 's' out of improvements and write: There is progress and improvement.

Only random person and stumbled have this correct. Very simply, the answer to your question is the verb are.

There are . . . .

1. Even though a singular word is closer to the verb, the verb is referring to 2 things: progress and improvement. Being close to the verb has nothing to do with this.

2. Often, something sounds right simply because we are accustomed to hearing it wrong. Example: "The data is correct" may sound right, but it isn't.

3. Whether or not "improvements" is plural is irrelevant. The verb is referring to two things. We use "are" when referring to more than one thing.

4. See #3. Also, the second example given as a correct version is a run-on sentence.

5. Yes.

6. That doesn't answer the question. The verb is/are is referring to two things (i.e., progress AND improvement). We use "are" when referring to more than one thing.

7. Yes. (Random Person has the right idea but for the wrong reason).

Q1: What "is there?"
A1: Progress and improvements

Q2: What is the cardinality of the answer?
A2: Greater than one

Answer to the original question:
"There are progress and improvements."

Of course, in Greek (even the bastardized version in use nowadays) this is not ambiguous to begin with.

-- Cheers

I would so not write that sentence, but if it were necessary to have it composed in that way, I would say "there is." Of course, that's because, if I'm forced into an awkward sentence, I want the reader to move over it as quickly as possible, and I perceive "there is" would cause less stumbling, not because I'm convinced it's grammatically exact. I can't find the reference right now, but I was looking this up recently and found that the rule is tending toward making the verb match the closer word. "There are improvements and progress" versus "There is progress and improvements."

Some examples, perhaps more obvious, that we expect "There is":

There is such love and joy in my heart.

There is enough time and money to go on vacation.

There is cheese and meat for sandwiches.

The source I was looking at said these were becoming correct because that's what we're coming to expect. Officially, the construction is still "informal," which brings me back to the idea that if that sentence is to be used in formal writing, I just would not use that construction. Others have made useful suggestions for the restructuring that I'll not repeat.