Question in English

I would say, "What time can we meet?"

Yes you can say that. You can also say what time can we meet?

they mean the same thing. but personally, for me, "when can we meet" is more general/vague. it can be answered with "one of these days" if the person does not really want to commit. meanwhile "at what time can we meet" is more direct because you're asking someone to choose the time/set the schedule. i guess you can use "at what time can we meet" if it's something that's important/urgent.

You can say it that way, but it's not really said. It's usually "When can we meet?" or "when are you free?"

what time can we meet

Yes you can I don't have any problems with that

Yes. "At what time can we meet?" and "When can we meet?" are both correct, though the latter might be asking for or expecting a rather broader answer, such as "On Thursday" or "Next week sometime" and you would have to narrow it down to a mutually suitable day or time. You would normally only use the former question if the day- and the decision to meet - had already been agreed. "When can we meet?" is an acceptable (and, indeed, usual) way of asking "Can we meet, and if so, when would be convenient?" "[At] what time can we meet?" is used when the agreement to meet (and the date) has already been established. Note that, although, "At what time ..." is correct grammar, very few English speakers would actually say that: most would leave out the 'at'!

The first one is what us Americans would say is broken English. Properly we would say “when can we meet?” Or “ what time can we get together?” Or even “what time works for you?”

I think that is also good term to use

I'm not an English teacher but yeah both can be used. "At what time can we meet" sounds like something I would say to my supervisor because it sounds like it is more urgent whereas "when can we meet" sound like something I would say to my friend as it sounds less committal and more casual. When are you free is even more casual.