Requests are ~て+ください, no を is needed or permitted.
Would the noun of the action tend to be omitted from this answer.
Both are allowed, but since the question is a very general 何をしていますか I'd favor including 戸 in this answer.
Am I correct in thinking that いる would be the better choice. Using 'to stand' is over the top because it is obvious I think.
I don't think you need either. All of your questions could be like 「Aさんは戸の前に、何をしていますか」.
Personally, when making flashcards of the sort you're talking about, I always take my examples from references. Textbooks, "A Dictionary of Basic Japanese Grammar", and Japanese-English dictionaries are all good, and of course anything that a native speaker like our Toritoribe-san gives you as an example is also good.
I particularly like the sentences from Shogakukan Progressive, which you can get results from online at goo,
NTTドコモが運営するポータルサイト「goo」。今話題のニュースやドコモのキャンペーン、サービス紹介などお役立ちコンテンツを展開しています。
Weblio has a much larger database of example sentences drawn from a variety of dictionaries and bilingual corpuses, but the quality of those sources is not always the best. The biggest problem to watch out for is sentences that only translate that way in a very specific context. Watch the attributions - if it's from a 辞典 or a 辞書 it's probably a fine example; if from a translation of a book, it has a high chance of being a very context specific translation.
英語例文の検索。ビジネスメールや英文の手紙、英会話などに。自己紹介やスピーチ、挨拶やお礼の文例もあり