Is It Safe To Take Flucloxacillin And Amoxycillin To Treat MRSA?

My son is on his second lot of flucoxacillin for MRSA and he had 5 days of co-amoxycillin after his first lot of flucoxacillin. His gastrostomy site is looking a lot better at the moment but still has some crusting around it. We are cleaning it every day. How long does MRSA take to clear up. Also I have quite bad eczema on my hands could I catch MRSA from his site?