This gets a bit complicated.
There are a number of verbs that express bodily or mental conditions, which hardly ever take continuous forms. I am cold, I have a cold/a headache/an itch/flu, I am in love, I'm tired/bored/excited are all examples. This has little to do with the meaning, because I am suffering from a cold/a headache/flu
and My leg is itching are normal, and the simple present cannot be used for them expect in a habitual sense.
So the answer to the first question is I've had a cold..., not I've been having a cold ....
Most of these can be used in the continuous, but it is comparatively unusual, and has a special meaning. I am being cold emphasises the continuing experience, and is usually conveying some extra experience or intention. It might imply I am enjoying being cold, or I am making a point of being cold so that somebody sees it. He's having a cold might even imply that he isn't in fact ill, but wants people to think that he is.
The specific implication of using this form might not be clear without context, but there will usually be some additional meaning being conveyed.
Feel seems to me to be able to go both ways. It can be used both in the simple form and in the continuous; and there might be an extra meaning conveyed by the continuous, but there doesn't have to be.
So I've felt this way ... and I've been feeling this way ... both work equally well; but as Armen says, in the negative this doesn't work, but that's not because of the word feel, but because haven't been Xing is incompatible with for ages (it works with a period of time which includes the present, such as recently or this month).