Hello,
what is this HA info page telling me?
HA is green and the peer is faulty. Does it mean, the peer is alive but unhealty or does it mean, peer is dead?

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You have a dead peer. I am sure you have tried to reboot, but I think you may need to break HA, check the 'dead' peer can still boot successfully, then re-establish HA
Faulty status is described in the documentation as below:

It could of course be external network factors, cables etc
Info for others: the next time we tried, the machine booted. Support asked me to check this BIOS settings with serial or KVM connection :
Check if the State After G3 is set to Power On. If not, set it to Power On.
In some BIOS option For State After G3 if you are getting option like S0 or S5 or Last state then here you need to select S0 state.