Por que Sam não tem que dizer "Aiya Eärendil Elenion Ancalima"?

Não precisa ser ativado pelas palavras. É ativado quando uma pessoa de forte espírito (e esperança) a empunha.

Visto no caso de Frodo quando ele se lembra do Frasco pela primeira vez:

'The star-glass?' muttered Frodo, as one answering out of sleep.... 'Why yes! Why had I forgotten it? A light when all other lights go out! And now indeed light alone can help us.'

Slowly his hand went to his bosom and he held aloft the Phial of Galadriel. For a moment it glimmered, faint as a rising star struggling in heavy earthward mists, and then as its power waxed, and hope grew in Frodo's mind, it began to burn, and kindled to a silver flame..., as though Eärendil had himself come down from the high sunset paths with the last Silmaril upon his brow. The darkness receded....

Frodo gazed in wonder at this marvellous gift that he had so long carried, not guessing its full worth and potency. Seldom had he remembered it on the road..., and never had he used it for fear of its revealing light. Aiya Eärendil Elenion Ancalima! he cried, and knew not what he had spoken; for it seemed that another voice spoke through his, clear, untroubled by the foul air of the pit.

^{_{The Lord of the Rings - The Two Towers: Chapter IX - Shelob's Lair}}

Frodo grita as palavras "Aiya Eärendil Elenion Ancalima!" automaticamente depois de o frasco está aceso. À medida que Frodo se torna mentalmente mais forte, o Frasco se torna mais poderoso.

A segunda vez que é 'ativada' é quando Sam a empunha durante sua luta com Shelob:

As if his indomitable spirit had set its potency in motion, the glass blazed suddenly like a white torch in his hand. It flamed like a star that leaping from the firmament sears the dark air with intolerable light.

^{_{The Lord of the Rings - The Two Towers: Chapter IX - Shelob's Lair}}

Para sua pergunta seguinte: Frodo não deixa cair o frasco no livro. Ele entrega para Sam, que mantém com ele.

‘That would not help us now,’ said Frodo. ‘Come! Let us see what Sting can do. It is an elven-blade. There were webs of horror in the dark ravines of Beleriand where it was forged. But you must be the guard and hold back the eyes. Here, take the star-glass. Do not be afraid. Hold it up and watch!’

[...]

‘Orcs!’ he muttered. ‘We’ll never rush it like this. There’s Orcs about, and worse than Orcs.’ Then returning quickly to his long habit of secrecy, he closed his hand about the precious Phial which he still bore. Red with his own living blood his hand shone for a moment, and then he thrust the revealing light deep into a pocket near his breast and drew his elven-cloak about him. Now he tried to quicken his pace. His master was gaining on him; already he was some twenty strides ahead, flitting on like a shadow; soon he would be lost to sight in that grey world.

^{_{The Lord of the Rings - The Two Towers: Chapter IX - Shelob's Lair}}

Sam então guarda o frasco no bolso, mas ele nunca o deixa cair.

Na adaptação cinematográfica de O Retorno do Rei no entanto, é mostrado que o frasco é ativado somente depois que Frodo canta as palavras.

Não canônico, é claro, mas fez uma cena muito boa.

Sam disse algumas palavras quando usou o Phial, cantando uma canção de Elbereth, em oposição à fala simples de Frodo que ele soltou. Na minha opinião, isso foi alimentado pelo amor de Sam pelos Elfos e seu conhecimento pelas músicas deles. Antes de usar o Phial, ele viu Elfos caminhando pelos corredores de Elrond e suas canções ganharam vida.

Galadriel!’ he said faintly, and then he heard voices far off but clear: the crying of the Elves as they walked under the stars in the beloved shadows of the Shire, and the music of the Elves as it came through his sleep in the Hall of Fire in the house of Elrond.

Gilthoniel A Elbereth!

And then his tongue was loosed and his voice cried in a language which he did not know:

A Elbereth Gilthoniel
o menel palan-diriel,
le nallon sí di’nguruthos!
A tiro nin, Fanuilos! ¹

And with that he staggered to his feet and was Samwise the hobbit, Hamfast’s son, again.
‘Now come, you filth!’ he cried. ‘You’ve hurt my master, you brute, and you’ll pay for it. We’re going on; but we’ll settle with you first. Come on, and taste it again!’
As if his indomitable spirit had set its potency in motion, the glass blazed suddenly like a white torch in his hand. It flamed like a star that leaping from the firmament sears the dark air with intolerable light.
_{The Two Towers: Book Four - Chapter X, The Choices of Master Samwise}

Uma coisa a notar é que não é certo que o seu Espírito indomável colocou o Phial em movimento, no entanto, acho crucial notar que a música de Sam reflete seus heroicos na história, e a opinião de Tolkien de que Sam era o verdadeiro herói da história.

Com relação ao seu segundo ponto sobre Frodo "soltar o frasco" depois de ficar paralisado. Isto está incorreto. Frodo entrega a Sam o frasco enquanto ele tenta cortar as teias de aranha com Sting.

Come! Let us see what Sting can do. It is an elven-blade. There were webs of horror in the dark ravines of Beleriand where it was forged. But you must be the guard and hold back the eyes. Here, take the star-glass. Do not be afraid. Hold it up and watch!
_{The Two Towers: Book Four - Chapter IX, Shelob's Lair}

[..]

Either she did not see Sam, or she avoided him for the moment as the bearer of the light...
_ibid.

Depois disso, Sam nunca devolve o Frasco a Frodo e, em vez disso, esconde a luz sob a capa élfica que recebeu em Lothlorien.

... and then he thrust the revealing light deep into a pocket near his breast and drew his elven-cloak about him.
_ibid.

[..]

Hardly had Sam hidden the light of the star-glass when she came.
_ibid.

¹Tradução da música para Varda:

O Elbereth Starkindler, from heaven gazing afar, to thee I cry now beneath the shadow of death! O look towards me, Everwhite!
_{Letters of J.R.R. Tolkien - Letter 211: To Rhona Beare}

É necessária citação, mas quando Frodo a usa pela primeira vez, ele está dentro do covil de Shelob (que pode ser um lugar de poder), não muito corajoso e longe da luz das estrelas.

Quando Sam pega, ele está fora do covil de shelob, e seu coração é corajoso.

Uma vez dentro de Mordor, o frasco não funciona mais, então talvez o covil do shelob seja como uma versão menos poderosa do Mordor, o que torna mais difícil acendê-lo.

(Eu sei que preciso de algumas citações para melhorar minhas respostas, mas não tenho o livro em mãos, desculpe)

De acordo com wiki, o frasco brilha quando Sam o usa em resposta ao seu espírito indomável. Você não precisa falar as palavras para ativá-lo; mesmo que Frodo gritasse as palavras, ele não fazia ideia do que elas queriam dizer.

As palavras Aiya Eärendil elenion ancalima! significa "Salve Eärendil, a mais brilhante das estrelas!"