Translation:In this picture, there is me and my friend Judy in front of my house.
I still haven't done what you want me to (and probably won't); but this time, I have put Judy before me because that is the customary order in English, while your sentence has the customary order in Arabic. As a translator, I believe my job is to make good, fluent English out of good, fluent Arabic. If I listened to you, my author-friends would no longer hire me to make good English out of their writing where necessary.
The word "there" being excluded from the sentence does not make for bad English grammar. This is especially true since the Arabic word for "there" is not in the Arabic sentence. While the word "there" is indeed optional here, the sentence should not be marked incorrect fot not using the "there" in front of "is" as either is grammatically correct. The purpose is to learn the trnslate good Arabic into good English.
Actually, not in this case. Indeed, according to The Cambridge Grammar of the English Language
A further complication arises in existentials when the verb is followed by an NP-coordination, as in There was/?were a bottle of wine and several glasses on the table. Were tends to be unidiomatic with an NP-coordination when the coordinate that is adjacent to it is singular, even though the coordinate as a whole (a bottle of wine and several glasses) is plural. Plural agreement, however, occurs readily in lists: There are still Brown, Jones, Nathan and Smith to interview.
In other words, the agreement of the verb to be after an existential there doesn't necessarily depends on the noun phrase that follows (in this case "me and my friend Judy"). It tends to agree with the closest part of the noun phrase (for instance "There is an orange and an apple", "There are some oranges and an apple", "There is an orange and some apples"), regardless of its status as singular or plural. Therefore, "...there is me and my friend Judy..." is not only correct, it's also the most idiomatic. I would be more concerned by the whole translation which is incorrect. It should just be "In this picture, my friend Judy and I are in front of the house"
This is still the best version of the sentence in English I've seen proposed.
I think the problem is that Arabic and English deal with the underlying proposition very differently, so that any attempt to stick closely to the Arabic syntax will result in unidiomatic English. Generally Duolingo makes that approach work pretty well, but with a sentence like this I think the only solution is to allow a wide range of different English translations — but of course that's problematic in itself. I suspect that we're up against one of the limitations of this sort of automated system, which requires a strict correct/incorrect binary, whereas a live teacher could use it as an opportunity to discuss and learn! It would be good if this discussion were a chance to do that, instead of sniping.
Having said that, the official translation is still not great.
Duo tranlsates to using using the woed "there" in the sentence when the actual word for there is not in the sentence commonly in these modules that are about position such as "in front of" or "next to", etc. But un this case putting the word "there" in front of "is" is marked incorrectly. This is inconsistent.
In English we wouldn't write "me" it should be "I", it sounds posh but that is correct. In spoken English we would say "me". Personally I would write this sentence without the "there" as well, you don't need it and it makes the sentence clunky. "In this picture, my friend Judy and I are infront of my house".