Sherlock uses the nicotine patches because most of the time he’s trying to quit smoking. In Hounds, it had been Sherlock’s idea to pay off everyone within a two-mile radius so they wouldn’t sell him any cigarettes. (Though he regretted it later when he wanted to smoke for lack of a case.)
The original character of Sherlock Holmes had a history of drug use. They haven’t directly stated it, but it’s been strongly implied that this version of Sherlock has a history of drug use as well.
Mycroft offering Sherlock a cigarette was evidently meant to be a test. When John spoke to Mycroft on the phone, he confirmed that it was a “danger night.” Then when Sherlock got home, he could tell John had been searching the flat (which he assumed included his sock drawer). I think the overall implication is that they were worried Sherlock’s accepting the cigarette was a sign that he might move on to other drugs.