Someone said to me that the first one is the correct answer, because you use "wake up" to ask someone to stop sleeping. But I think the first one does not sound natural.
In modern (American) English, "wake" in your example is generally followed by "up". But this isn't always the case.
I don't always like to use NGrams but here's a helpful image that shows the recent trend:
Of course, this misses some various other phrases that are similar, like the command "wake me at 7 am"... which is fine but can also be phrased "wake me up at 7 am".'
So, to answer your question, in modern English, "wake up at" is more appropriate and common but "wake at" is not technically incorrect... it's simply not much in modern use and you may find pockets where it is still quite acceptable.
I believe it's the second sentence that's correct and that your friend's misinformed. One could say Are you awake at seven? but omitting up sounds like wake is something one does when one watches over someone.
According to the wisdom of the net, there are other meanings to the very to wake and those use different prepositions. However, the act of entering a conscious state interrupting ones sleep is usually accompanied with up.
Usually, 'wake' is followed by 'up' when you talk about getting up from a sleep or a nap. And I don't think 'Do you wake at seven?' is the correct way to ask that. Technically, it is not wrong, but ideally, the latter is a better option. Generally, you 'wake' and you 'get up', which is exactly what 'wake up' means.