May one gather eggs from a chicken coop on Shabbos? 
It is forbidden to be done even if the eggs were laid before Shabbos and are not Muktzah as gathering items that do not grow from the ground.
However if delaying their gathering will cause a great loss one may be lenient to ask a gentile to gather them.
May one make a necklace or bracelet by entering items into them?
This is forbidden due to the prohibition against fixing a vessel, and according to others also due to the “Meameir” prohibition. Some however permit it to be done, as they say that the Meameir prohibition only applies by gathering items that grow from the ground, and they do not hold that doing so involves the fixing prohibition.
May one make a flower bouquet on Shabbos?
 Ketzos Hashulchan 146 footnote 49-22.
 As gathering items that do not grow from the ground in their area of growth [which in this case is the coop] is Rabbinically forbidden.
 Sheivet Haleivi 4/39 As is always the ruling regarding asking a gentile to do a Rabbinical prohibition in a time of great need.
 Ketzos Hashulchan 146/footnote 49-25; Piskeiy Teshuvos 340/26
 Ketzos Hashulchan 146/footnote 49-25
 Orchos Chayim 13
 Shevisas Hashabbos Meameir
 According to all it does not involve the writing prohibition as there is no prohibition in forming a word as explained above.
 Igros Moshe 4/73
 This is forbidden due to fixing a vessel, as the gathering of the different flowers to make a bouquet are considered making a nice vessel out of the flowers. [ibid] However see Ketzos Hashulchan 8 p. 92 which seems to imply that the only question involved in gathering the flowers together is the prohibition of Miameir. However when done outside of the area of growth, such as in one’s home, it is permitted. Hence implying doing so does not carry the Tikkun Keli prohibition. Vetzaruch Iyun on the essence of the logic itself to consider simply placing flowers near each other as Tikkun Keli. It does not appear at all like one is making a Keli. This is unlike a necklace which when one enters the beads it is clearly apparent as if one is making a Keli. Vetzaruch Iyun.