Hi Hun. My lo was very ill at 2 weeks old and was treated for suspected meningitis just like your boy. He had a lumbar puncture and the same broad spectrum of antibiotics (one is amoxicillin). He was also having calpol because of his fever. He had the rash, spots etc. it turned out he had a blood infection. Still serious but he's fine now.
My oh at 1 year old had meningococcal meningitis. It was affected his brain and speech. However this was back in 1985 and medicine has come on loads.
Lots of hugs, hoping your lo responds to the antibiotics and he's home soon.