Opinions on Alcina? Not Alicina, not Alicia, not Alice. It has nothing to do with the name Alice actually. It means strength or defense.
Alcina is the character of a somehow famous Opera and a main character of a book I really liked. I pronounce it al-CHEE-nah, and that is probably how it will be pronounced where I live anyways.
Alcina is also the sorceress of Orlando Furioso - I am not sure how much it is known as a book where you live but it is one of the major literary pieces of my country (I feel?) and it is very well known here. It was written during the Renaissance around and I love the literary yet Renaissance vibe, but still able to feel vintage and 19th century-ish (at least in my context).
I am struggling in finding diminutives and nicknames too. Can someone think anything for me other than Alchie/Alci?
It’s not my thing, but I don’t like many (if any, can’t think off the top of my head) -ina names in general. I understand it’s appeal for the same reason I don’t like it, I find it quite frilly-feminine in a similar vein to [name_f]Alicia[/name_f] etc, visually very similar. I do love the literary associations you’ve cited and can see why it appeals for this reason also.
Other nns - [name_f]Ally[/name_f], Cina (prn. Cheena, couldn’t spell it [name_f]China[/name_f] for obvious reasons).
[name_u]Ali[/name_u], [name_f]Nina[/name_f] and [name_f]Lina[/name_f] would be my go-to nicknames! [name_f]Cia[/name_f] could work too. I’d pronounce [name_f]Alcina[/name_f] the same way you do at first glance. Personally, I don’t love the sound, but I don’t really like [name_m]Al[/name_m]- names in general so it’s a ‘me’ problem 
I love [name_f]Alcina[/name_f] it was on my list once
I would use nn [name_u]Ali[/name_u] or maybe [name_f]Ana[/name_f] or [name_f]Lina[/name_f].
[name_f]Alcina[/name_f] is intriguing and it does have a literary, magical vibe. It’s sleek with a familiar but distinctive feel.
Potential nicknames:
[name_u]Allie[/name_u]
[name_m]Archie[/name_m]?
Chee
[name_f]Ina[/name_f]
[name_f]Chia[/name_f]
[name_f]Cece[/name_f]