Yes. And note that this is true even for just a single person's utilities, i.e., without even getting into the issues of interpersonal comparison. For example, a single person, just to compute their own overall utility (never mind taking into account other people's), has to be able to aggregate their utilities for different things.
> if both exist, then there is a representation where utilities can be represented as (a subset of) the reals.
Yes. In more technical language, total ordering plus an aggregation function means utilities have to be an ordered field, and for any reasonable treatment that field has to have the least upper bound property (i.e., any sequence of members of the field has to have a least upper bound that is also in the field), and the reals are the only set that satisfies those properties.
That also raises a different question about people's introspection and ability to accurately answer questions about their utility and preferences, but that's a bit far afield from the mathematical question.