Often heard after exposing some matters in front of a class:

Questions ? Commentaires ? Discussions ?

It is an invitation to ask questions, requests details, expose your comprehension when you doubt you understood everything, etc.

It would be used in pretty much the same context as your “Any questions? Any queries?”.

I am not overly familiar with databases, especially in French, but I believe the English query is usually translated as requête…

Just for adding informations about the database part. In France, we are used to say :

Le programme interroge/questionne la base de données.

Or

Le programme lance/envoie/fait/crée une requête à la base de données.

Even if the verb “to request” exists in French, it is never used in this context.

Both above solutions are commonly used. The first one will be used when you want to say that your program query the database (global behavior of your program). The second one is more focused on the query itself.

There are the words question, requête and demande in French.

Question is when you ask for information, about something you don’t know.

Requête is used in administrative or legal context, when you ask for something to be done (instead of asking for information). It can also be a formal way to ask something to do from somone.
It is also used for databases as the translation of query.

Demande is more or less the same as requête but is less formal, it is also less authoritarian because requête can be used for conflicts (and yet be less authoritarian than a demand) while demande[Fr] would be used for negociation or a genuine request.

According to your description (“query” is used in the context of a need of clarifying a doubt, expressing a doubt or getting information about something), a query might be une interrogation (doubt) or une demande (getting information about something) while “question” would keep the same meaning in both languages.

You might then ask after a meeting:

Des questions ? des interrogations ?