Yeah I think "bei" would be sounding like someone is near to the door, not at the door. Could say "neben" here, but that sounds like someone is next to the door, whereas with "bei" one could mean they are slightly to this side of the door or something like that. German is very precise.
As the above examples show it can be used in some situations, in some specific contexts. But you're right that that "There's someone at the door" is more often used and really a better translation than a literal one.
Es gibt/Es ist/Es sind are equivalent to there is/there are, and it would be better to memorize them, instead of looking for literal translations.
fjt30trad, e.brown and Dodoasadi, if my comment above is correct, as Tür is a feminine noun and as that "an" requires dativ, it must me "an der Tür". You can see the table of cases here: https://docs.google.com/document/d/1zeWbBQ4r1bKILt_e-FXCrzXl6d1ru2OVtui2c82pyuM/edit#heading=h.z1xlvfchb13h (it was posted by the user _wednesday in another post) There are 2 problems to solve this sentence: 1) Gender of the noun - Tür: feminine 2) Case: "an" is a two way preposition, it can lead to either accusative or dative (in the same link above there is a list of two-way prepositions, however, you can see a more complete explanation here: http://german.about.com/library/weekly/aa052101a.htm) 2.1: Accusative in two-way prepositions is normally used in sentences in which the object is in motion 2.2 Dative: object is not in motion, thus there is a location determined. The example used in one of these links is the same idea of "in" and "into" in English
Therefore: feminine + dative : "an der Tür".
Please, someone could confirm the statements above? I hope I have helped.
Apache. You are completely correct. Just to clarify somethings: "an" is used for vertical surfaces (e.g. walls, doors, gates) and "auf" is used for horizonal surfaces (e.g. beds, floors, couches).
Er läuft an die Tür = He is walking up to the door. Ich warf den Spiegel auf das Sofa. = I threw the mirror onto the sofa Accusative; Motion;
Er ist an der Tür = He is at the door. Ich legte den Spiegel auf dem Sofa. = I laid the mirror on the sofa. Dative; Location
Absolutely, but the resulting English sentence fails to make sense in this context. I would think of es ist here like es gibt. As in not the direct translation, but remember how it's used.
Since I'm not a native German speaker, I couldn't really tell you the difference, but as I understand it, "Da ist jemand an der Tür.", "Es gibt jemand an der Tür." and "Es ist jemand an der Tür." would all translate to the same thing. "There is someone at the door."