"I love my cooking fish(Here my is unnecessary?)"
Depending on the intended meaning, yes and no.
You could re-write without 'my' like follows:
I love cooking fish
This means the person simply loves cooking fish (the fish could belong to the person, or the fish may not belong to them).
If you wanted to use 'my' to signify that the fish belonged to the person cooking them you would have to change the word order like this:
I love cooking my fish
"Besides I love my feelings
(Here is ok, because means something you have got but not an act)"
As this stands it does not make sense. If it was part of some other sentence you could re-write it like this:
Besides, I love my feelings.
The word 'besides' is standing in relationship to something else, like this
Y besides X ,
e.g. I don't go the gym, besides, I don't need to I'm not unfit.
Here we are using besides as a preposition, where it means 'over and above'
But there are two [words] in dictionary confuse me:
Greeting: something you say or do to greet people(It is like the example "feeling"?)
It could be or it may not be. It depends if you use it as a noun or a verb.
'A greeting from an old friend' here it is a noun. So similar in use
to the 'my feelings' example.
'I am greeting a friend' here it is a verb.
Coming: the time when something new begins(It is like the word "cooking" or "swimming"? Because it has some verbal characteristics)
It entirely depends on how you are using the words.
In the sentence "I am coming home"
the word coming is similar to swimming "I am swimming" in the type of word it is.
It is something you are doing, a verb.
The verb is 'come' but when you use it like you have here, ending in 'ing' you are using the present continuous tense.
This means you are doing something now, in this moment.
Cook --> Cooking
Feel --> Feeling
Greet --> Greeting
You can cook something.
You can feel an emotion.
You can greet someone.
You can cook something 'now' - in which case you are 'cooking
something' You can feel something 'now' - in which case you are
'feeling something' You can greet someone 'now' - in which case
you are 'greeting someone'
As a general rule of thumb, when you see a word ending in 'ing' you can think 'this is happening now' there are exceptions of course, but hopefully that helps.