Some scenarios:

1) Removing bones in chicken or fish? 

Can't do unless take some of good with it or whilst eating it to pull the bones out

2) Removing skin of chicken?

Ok as considered same "type".

3) Trimming fat off meat?

Must trim the meat from the fat.

4) Peeling an egg?

This is fine as long as by hand and just before the meal.

5) Cracking a nut?

Also fine.

6) Discovered small insects on lettuce how to dispose of the insect?

Can't rinse leaf as that removing the bad from the good.

So, solution is to cut away and discard part of leaf together with the insect.

7) Using a slotted spoon to remove salad or chullent?

This is like a strainer ao no good. Unless it's a soupy chullent or thick dressing and undesired liquid isn't intended to be drained off.

If you don't want the liquid drained, it's fine anyway.

8) Using a pot lid to allow soup to pour out whilst the noodles stay inside?

This is a problem unless he makes sure to leave soup with the noodles.

9) Organising pieces of a game of different colours and returning to different compartments after been used?

No good

10) Putting assorted books away that are left piled up?

Best solution maybe to read a bit from each book before you put it away.

11) Setting the table on shabbos?

Caution to do close to the meal and taking wanted cutlery from others not wanted.

If it gives a nice feeling to have dining room set, this could be considered just before use even if done hours earlier.