No. <- that
I know that in the books it is implied that Ciri has feelings for Geralt, but those aren't mutual.
Where? Only thing of that sort that I remember was when little Ciri was jealous when she met Yennefer. So far Geralt was only for her, and now new woman appears which can take some of Geralts attention from her. But it was about jealous daughter who doesn't like new stepmother kinda feeling, not love towards Geralt with sexual context. It was brief, not sexual, and unimportant. Grown up Swallow is fully aware and have nothing against Yen and White Wolf relationship, which is best shown in the bathtub scene at the end of Lady of the lake.
So in TW3III there is no place even for "rejected affection"
That being said, I believe that Ciri has what it takes.
She doesn't need to be put through mutations because she was already born with powers.
Yes she is conclusion of controlled and uncontrolled experiment spread over generations. In some way she deserves to be call "mutant" more than witchers are. But, Witchers are mutated in very precise, calculated and purposeful way. To fulfill very specific and demanding stuff so you can't put equals sign between her powers and witchers mutations.
-she has a witcher medallion
She took it from a dead body, didn't earn it. It was very symbolic scene, meaning that she is a complete warrior now, able to do amazing stuff like reflect arrow with her sword for example. But not a witcher.
-she knows magic much stronger than signs
She is able (assuming, that her "rejection of magic, was purely mental blockade - and I agree with that version)
to learn much stronger magic than signs (but so are witchers as Vilgefortz explained) . As far as we know she know very basic stuff, that she learned from Yen. She never completed her magic education.
-she defeated master swordsmen (i.e. Bonhart, Eredin)
Yes, she did. She is outstanding fencer. But is she on witcher level? It's debatable. On one hand we have her reflecting an arrow with her sword (ability associated only with witchers), killing Bonhart (which was not luck but pure skill, and ability to take advantage from better position). On the other hand, we have Coën who says that she will never be as fast as witcher, and Geralt, who instruct and corrects her after seeing her defeating few men (near the end of saga).
-she learnt all about monsters in Kaer Morhen (she had lessons)
She took lessons. We have no idea, if she mastered all of witcher knowledge. Personally I doubt that.
-she ran The Killer daily at Kaer Morhen and bested it like other witchers with mutations did
So? Was it such an achievement, to learn how to control your breath?
...the only thing she misses are the mutations, which Geralt wanted to put her through
This and all above..
But it was prevented by a certain sorceress because it would have made Ciri infertile and foiled
the lodges plans regarding the Lara Dorren gene.
It wasn't "prevented". Witcher never wanted to put her through trials in the first place. Triss theorizes about that during her journey to Kaer Morhen, but quickly abandon that idea (as she was called to temper Ciris "prophetic trends").
So ....yeah. To me Ciri is a witcher, although a very unexperienced one.
But I have no doubt that Geralt will vesemir her to his best abilities ~*
No she isn't. No trials, no witcher specific mutations, no monster slaying experience, not enough training.
I just have one thing to add:
Steel swords can't cut through heavy armors, chainmails.
At best they dent the armor which causes severe injuries to the person wearing it.
But that is irrelevant because a witchers fighting style does not aim at armor, but uses precise strikes
against weak spots such as arteries, the lung or the heart.
Witcher do block with sword. Silver one isn't exactly best suited to do so. Silver is good against monsters, not other fencer.
If someone comes wearing an armor they get greeted by an Igni and boiled like a sausage.
Igni from games, maybe, igni from books not so much. Besides, Ciri didn't know how to create and use igni.