I just looked up some stuff on the HTC website. They don't have any support in Sri Lanka, which is an issue for you.
If you've got any relatives or anything abroad? Because it's impossible to get it fixed in your country.
One of the only solutions I can think of is ship it off to singapore repair services, but that's expensive and not worth the money most of the time. Your dad is probably better off buying a new HTC desire (normal one, the Z which I have or the HD)
Also, if you think your English is good enough, you can ring a international HTC number (comes with high fees though) and ask them what you need to do. You can find a list here
Wow. First off, you don't need to root your phone to install free apps (you're thinking of the Fruity phone), and you don't need to root your phone to reflash the firmware either. You can connect it to the PC with the official software and reflash the official firmware without rooting.
Anyway, the OP's problem could probably be solved by factory-resetting the phone. Make sure you have all your contacts backed up to Google's server. You are probably going to lose most of your game saves unless you root the phone and back up the game files though. If you want to do that just you need to root your phone and then using an app like Titanium backup to backup your game data. If you don't care about that you can just factory reset your phone by going to Settings > Privacy > Factory Reset.
Have you got any relatives abroad who you trust? HTC provides no support in Sri Lanka, so you can't send it in for repairs.
I wouldn't go around opening it up, because that's risky, and voids your warranty. Contact HTC on one of their numbers, in whichever country is closest to you or ship it to a relative, who can then ship it to HTC to get it repaired.
I'm not the most experienced with troubleshooting smartphones, but random restarts sounds like (to me at least) a problem with the power supply (i.e. the battery).
As a diagnostic step, try taking out the battery and checking the contacts (on the battery and in the phone) for any signs of dirt or foreign objects (e.g. hair) that might be obstructing it.
Also, if you do choose to send it off, try two things before you do. First, take it to a trusted phone shop and ask for their opinions/advice. They're more likely than random forum users to have encountered this problem before, and may know how to fix it. Second, try calling HTC support and seeing what they have to say. It's probably a common problem with a simple fix, or maybe just a very basic hardware fault that takes minutes to fix. They'll probably know best on what course of action is best.
EDIT: Oh, and remember to take out the SIM/SD cards. You won't need to send them in with the phone, since they'll use their own tools.