I have two sets of instructions for putting the wort into the primary fermentation bucket. One says to just dump the entire contents of the wort into the bucket. The second says to stir like crazy to create a 'hop cone', then syphon the wort, minus the hop cone, into the bucket ...
Oh, what to do????
My protocol is to chill the wort in the kettle (I use an immersion chiller), then "stir it like crazy" -- it's called whirlpooling -- then wait at least an hour (I usually brew after dark, so I leave the covered kettle overnight) and siphon the wort into the fermenter.
This helps to remove both hops and hot&cold break material, which may give off-flavours to beer.
Straining hops from pellets is a nightmare, and I wouldn't advise that to anyone. I usually use hop bags, anyway, so I whirlpool&siphon only to get rid of break material.
I used to just discard the remaining ~1 l (in my setup) of liquid in the kettle, but last time I strained it, gave it a quick boil and added to the fermenter. Some people actually save that portion for later krauzening.
Putting hops in the fermenter after fermentation is called dry-hopping. Some brewers do this to get more hop aroma. (See https://en.wikipedia.org/wiki/Hops#Brewing) However, that would be with new hops, not the ones you already boiled.
It's best to remove the hops, but it's not a deal breaker if you don't. It will just make it harder to siphon the beer later. There are 2 other options to consider...you can put the hops in a nylon or muslin bag so that the entire bag can be removed later. Or after the wort has been cooled post boil, you can pour it through a sanitized strainer into your fermenter to remove the hops.
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