Hello, I come with a really strange error.

I noticed the trouble when navigating with firefox. It randomly broke tabs, appearing the 'some error occured' tab. Then when I do md5sum of a large file, it is different each time. When burning a live usb stick of some distro using dd, at boot time, some random error occurs.

To do more tests, I booted a linux live image (created in another machine), and did:
dd if=/dev/zero of=./file

It copies to file as zeros as possible for the media. Doing a hexdump to this file, gives some 1s at random positions, different each time hexdump is executed. However, doing hexdump to /dev/zero, does not gives any 1.

I have no idea what is happening here.

Thanks

The disk failed. If it's a physical disk, run fsck on it; if it's flash run f3write/f3read on it.

Check the cable as well.
You wouldn't believe how often I've been asked to change a faulty disk that just wasn't plugged in correctly or had a faulty cable. SATA connectors are FAR too easy to pull half off. That is the one big design error with SATA.

From the described symptoms, I would first suspect a memory problem. Boot from some installation or rescue medium that offers a memory test and let it run for several complete passes.