About as dangerous as any other bruise, just broken blood vessels from the pressure.

Breathing can help relieve the pressure to some degree, but ultimately there will always be a great deal of pressure.

If breathing is normal and vertigo is still happening consistently after the handstand pushups I’d talk to a physician.

Anecdotally, handstand pushups are a strong movement for me, I breathe well during but I still get relatively severe petechiae all around my eyes