Can two applications communicate on an Ad-Hoc network?

Regardless of the topology of your network, direct access from one PC to another will depend on your ability to address it , either via IP address, DNS or whatever ( For example, in your ad-hoc network computers have names ? You can simply use that name to get the connection - just as you already do through the file system.)

If all computers are "on the same level" then it should be possible to access via IP. Use ipconfig (windows) or ifconfig (linux) on the computer where the database is, and look for its IP address (if there is more than one, what interests you must be the local network, and 192.168.X.Y - where X is usually 0 and Y is greater than 1 ). Try to put your DBMS listening at this address, and on the other computer try to access it by that address [plus the correct port, if necessary].

Now, if a computer is on a subnet and what it is wanting to access is outside of it, then it is necessary to configure its own routing or maybe a proxy. In this case, I do not know enough to give an opinion.

If the application works via the network, no matter what the structure of the application.

If you have IP on both hosts the application should work (taking into account that the firewall is not blocking communication).

Yes, since you do not need the Internet or one of the hosts acts as a gateway ...