The settings on this computer do not match the requirements...

I haven't tried with Safe Mode.

I rebooted the router and now the internet works. This once happened before but still turns back to the X sign after a while. I hope this time it doesn't.

Do you know if they added a new device while switching ISP?

Ohh, the X sign just showed up again. -___-
Added a new device while switching ISP? I don't have any idea.

Well, could you please explain to me why it takes time before the X sign shows up and connection goes off?
I just wonder that, if the settings either on the router or laptop are not correct, I wouldn't have been able to connect in the first try. But in fact, I was able for a moment before connection goes off. Moreover, I am still able to get online even if the X sign is showing (like this moment). Though it will surely go offline later.

If this was only one PC we could try some PC specific networking fixes but I feel it is useless at the moment. Please try with Linux Mint MATE: download and burn the ISO to a USB flash drive with Rufus or to a DVD. Boot with the USB/DVD and see if network connection will be stable.