Past tenses change not only according to grammatical gender but also to actual gender.
If the subject is female, it's takes the feminine form. If the subject is male, it takes the masculine form.
Either one should work here since there's nothing that tells us the gender of the person speaking.
It's fine - but uncommon. In the UK and most parts of the US "to wait on someone" is understood as "to serve someone". (I think the Russian term for that is "обслу́живать" but I'm not sure.) However, in some parts of the U.S. (mainly in the South) "wait on" is commonly used as a synonym for "wait for" so I'd argue it's an acceptable translation (at least in Texas...). Although, personally, I wouldn't use "wait on" in place of "wait for". "Wait for" is less ambiguous and much more widely used.
In another exercise - Подожди автобуса - there was a lot of comment about this question. What I gathered from the comments was that, with ждать / подождать, inanimate things waited for are usually Genitive, animate beings are usually Accusative, vehicles can be either. Messages are usually Genitive, unless they are definite, e.g., about to arrive.
егo can be either Accusative or Genitive, so there's no way of telling from this exercise what it is, but the comments indicate that it's Accusative.
In English, "waiting on" someone means to act as an attendant, and to perform tasks for that person. At restaurants, the staff "waits on" customers - thus they are called "waiters". A restaurant is the usual place where a person is "waited on", but that service can extend to other places and activities. There is an English idiom, "I waited on him/her hand and foot", which means "I performed a lot of services for him/her, at his/her command and orders"
Depending on where you are from, some folks in America say they are "waiting on" someone to indicate they are waiting for someone. It may not be widely used, but I have heard many people use this in my life. So for some, that could indeed be an accurate translation of colloquial speech.