Further investigation tells me that "d'à côté" is used to mean "next door", but it seems only in the adjectival sense - "la maison d'à côté" = the house next door; "les voisins d'à côté" - the next-door neighbours. But I haven't found it standing alone, as in "we are next door."
Francophones? Au secours?
UK English uses "next-door" as a an adjective translation of "d'à côté", as DianaM has pointed out. Similar expressions include il habite à côté = he lives nearby ou close by. Given the closeness of these terms, I'm sure you would be understood in general if not literally about about next door. If you had some context (e.g., chez vous), it would be quite reasonable to say "next door".
Presumably you meant between "sommes" and "à". There are a myriad rules about when liaison is required, when it is forbidden, and when it is optional. Plus there are degrees of "optional" - i.e., some are more commonly heard than others. Sigh.
I couldn't find a rule that specifically covers forms of être followed by a preposition, but perhaps Section III on this page covers that, too. http://french.about.com/library/pronunciation/bl-liaisons-o.htm
I totally understand. There are those who prefer a literal (but awkward) English translation just for the purpose of enhancing understanding and facilitating the reverse translation. IMO, that is an over-simplified approach to language learning. If you were staying with a family in France for three months, I don't think they are going to use dumbed-down French. Good French (correct and natural) should be translated to good English (correct and natural). So, hang in there, Lisa!
'alongside you' is perfectly good English - in my part of England, anyway! It should certainly be accepted and it is perverse that it is not. I have reported this before...