You might want to look at the
Uninstall Problems thread for some suggestions. If you want to remove mIRC from your computer you need to go through 'Add\Remove programs', not just deleting the .exe file - although deleting the .exe file should not mean that it is re-created on reboot.
As you say, it is most likely a trojan causing the issues. Take a look at the
Antivirus and firewall resources thread and use 2-3 virus/trojan scanners. I recommend AVG and The Cleaner personally. Also check for spyware using a couple of the programs listed in that thread. I would also recommend installing a Firewall. You should always have antivirus software on yoru computer - if applicable, 'auto protect' features should be enabled for round-the-clock protection. You should regularly check for updates too - an out-dated antivirus is useless.
G'luck!
Regards,