Learn English – Differences between “Pay attention to” and “Keep an eye on”

An incident is a single distinct event - one occurrence of something that can happen more than once. It is often unpleasant and/or unexpected.

Issue has lots of meanings. Given the other words you're asking about, the meanings you might be thinking of are (1) a problem, something that needs to be discussed/fixed; or (2) a topic of interest.

The first meaning that springs to mind for affair is the extra-marital kind, but it can also mean a social event (of the dress-up kind), or a plain old (read: not sex related) scandal.

(I've linked each word to the definition with the most examples of usage that I could find.)

After that incident with the neighbor's dog, he avoided chihuahuas like the plague.

vs.

After that affair with the neighbor's dog, he avoided chihuahuas like the plague.

The incident probably took less time than the affair - for the former, I'd imagine the dog scared the man, while the latter might have involved an actual bite and subsequent lawyers and suits and animosity.

After that issue with the neighbor's dog, he avoided chihuahuas like the plague.

Here the implication is that there was a problem which had (or should have had) a solution. I'd imagine the dog peed somewhere inappropriate, or something similar.

As far as I can tell, pay is used when there is no possessive adjective (e.g. my, his, etc) before the attention:

I didn't pay him much attention.

Give is used when there is such an adjective:

All of us were supposed to give Chris our undivided attention.

It is interesting that give attention (without the possessive) hasn't always been uncommon, though it's always lagged behind by a factor of at least two:

As you see, pay attention currently leads by ten times or more.