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.
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"