What's the best way to determine whether Firebird server is running?

If you want to do it from an application, a simple try to connect should suffice. Otherwise you have various options:

a) check if firebird server is in the list of running programs (use task manager on Windows, or 'ps ax' command on Linux). Please note that Classic won't be running until there is a connection established.

b) check whether the port 3050 is open on the machine. First, you can check with netstat command, and if it is open, you can test whether it accepts connections by telnet-ing to the port. Just type:

telnet [hostname|IPaddress] 3050

Example:
telnet localhost 3050

If you use Linux, you can also check the open port with 'lsof' command. It outputs a lot, so you might want to 'grep' for 3050 or gds_db strings:

# lsof | grep gds_db
# lsof | grep 3050


c) if all of this fails, perhaps you should check whether the remote server is reachable at all. You can use 'ping' command:

ping [hostname|IPaddress]

Example:
ping 192.168.0.22

Please note that ping can still give you 'host unreachable' message even if host is up. This is because the firewall software can drop the ICMP (ping) packets (it's done to prevent some viruses from spreading, or network scans).