Can a defective or shorted mouse cause my motherboard to burn?

Let's go by the numbers, so to speak:

1) does the motherboard feed pilot LED comes on when you connect the PSU to a power socket?

If so, power still goes in the motherboard.

2) most modern motherboards have built in safeguards.

Your motherboard being ready for overclocking is even more likely to have it. Your PSU is still salvageable , the CPU and HDD most probably are too; if you have an SSD, the risk it may have got corrupted or even burn goes a bit higher. RAM is a bit of a hard thing to put a finger on. It may be the case the sockets went bust but the chips itself are functional; test it in another machine if you can.

3) your machine not booting may be caused by having the USB socket shorted.

I once had a case of a desktop that would not boot because of a mouse that shorted the USB where it was connected (although not with a as dramatic outcome as yours) and the solution for the problem was to simply replace the mouse. So, yes, your machine may appear dead because there is a safeguard in place preventing it to power up and fry the rest of the machine.

4) what else was living inside your machine? Graphics cards, video/TV or other type of expansion cards? Those are more vulnerable to short circuits than anything else.

5) replacing USB ports on a motherboard may prove challenging.

The backports require a bit of skill and ingenuity yo replace but it is doable. If you want to do it, you'll need to post here more pictures of the affected area; the soldering points may be in plain sight or be hidden and require more work to reach or may even be inaccessible without specialized tools.

P.S.: you can salvage parts from one machine to the next around 95% of times. Unless you have a catastrophic failure - epic power surge. like a lightning strike, or a fire starting from the PSU - some part will survive. You just have to have the patience to go through it and test everything.

P.S. 2: ASRock is an offshoot of ASUS, originally intended to supply the entry level budget consumer market. Now, it rivals its parent company and supplies the full range of consumer motherboards, from the entry level to the more high end motherboards, for enthusiasts and gamers. There's a lot of trash talk on the inter-webs about the brand but so there is about any other brand; don't allow yourself to be fooled by angry fan boys (oh, how much I wanted to use another expression regarding the fan boys...).

Don't consider your machine a total loss as it is now.