Sherlock did not sleep.
'Sleep' was too tame a word for whatever it was Sherlock did every few days (or at the end of each long case) in order to recharge. He would call it 'sleep', and he would follow a similar routine that any other human might, but John felt that the act of lying comatose for (sometimes up to) 20 hours and being oblivious to everything short of a bomb detonating in the kitchen should probably be considered as more akin to "power hibernating" than to sleep.
It wasn't even as if it was a restful time for the consulting detective; John couldn't count on both his hands (and feet) the number of occasions upon which he had rushed into Sherlock's room at the sound of a gunshot to find that his flatmate had, in his 'sleep', simply thrown himself across the room and ripped the curtains down as if they had personally offended him, or, as had happened recently, thrust his antique rapier blade-first into the side of his bed frame.
When he wasn't experiencing bouts of sleep-violence, Sherlock was usually holding loud debates with himself about the merits of Tarmac versus concrete or why accents are different in the north of England compared to the south or even why cats don't meow to communicate with each other and frankly John thought it couldn't be very replenishing for the man at all.
(Those thoughts often disappeared during cases, when they were three days in and John felt almost hungover from exhaustion and Sherlock was still as energetic as ever, running from flat to Bart's and back like a greyhound on a caffeine high, his mind working even faster. To be perfectly honest, it drove John spare.)
Sherlock did not have an eating problem, per se.
Obviously, there were times during cases where John had to hold the man down and force feed him (which was sometimes difficult, surprisingly, considering the fact that Sherlock spent most of his time sat on his backside) but outside of cases, Sherlock had a decent sized appetite.
John often thought that Sherlock was at his most irritating (and, admittedly, amusing) when he was mooching about in the kitchen cupboards in search of food between the body parts and petri dishes (pulling disgusted faces at the vegetables that John had actually hidden so as not to offend him) and desperately trying to find a Snickers bar that didn't predate their tenancy.