Dangers of Public Computers, How Likely Is This To Occur? Help...

Hey everyone, in my free time I like to play video games such as Valorant, League of Legends, or Overwatch and I have always played it on my personal computer in my room. I recently began university and my school has a nice lounge (we are sponsored by companies like alienware and logitech) where we have things like a switch setup, about 12 PCs, and XBOX/PS5's.

The thing is, i've never played on public computers before but I was always told by my SIBLING to never do so because of 3 concerns which I am not sure are completely reliable..

  1. He says if I log into any accounts on a public PC at the lounge (battle.net account, riot account, etc) then if there is a keylogger or malware the person downloaded from before me, I could possibly lose my account.

  2. He says that if I plug any of my own peripherals like a mouse or keyboard into a public PC, then if that public PC has malware/viruses then it could get into my mouse/keyboard because my peripherals are razer products that have software on them.

  3. *Important* He says that if someone hacked my gaming (riot/blizzard) account while I used a public PC, then if I logged into that SAME account on my personal computer that was hacked then I could get hacked too, since the hacker could put files onto my gaming account and then when I log into my personal computer with that same account it would get these files on my computer.

AGAIN, this is just what I have been told but I am not entirely sure because I am not very tech savvy, I just enjoy video games. But some of this stuff seems like a stretch so I wanted to know if anyone could give any input!