Regarding the coat--I do think there's an awkward little dance that sometimes happens, when you're trying to be generous but not pushy, and sometimes we just don't get it right. A lot of the people I know genuinely forget they wanted to borrow something, or loan something, so I often feel like I have to push just a smidge and be like, "So, did you want to borrow the thing/loan me the thing, or not?" And often the response is an enthusiastic, "Yes, I totally forgot! Let's take care of that."
I like to make the "or not" explicit somehow--"or did you change your mind/do you still need it/are you still loaning it out" etc.. Give them a face-saving reason they can grab onto if they've changed their mind (or never wanted to in the first place), and force them to actively agree if they do still want it.
Regarding food--I usually say, "No thanks" once, and then if pushed a second time, I say something about my restricted diet. On the one hand I don't really like discussing personal stuff like that, especially because people often ask follow-up questions at that point; but on the other hand, it seems to make the whole experience more awkward if I don't tell them I can't eat the food until several steps in. Then it's kind of like, "Why didn't you just tell me that at the beginning?" and I'm like, "Why didn't you just take my first 'no' seriously?"