"Tha i ceart", well, sboso. Literally, "It (or she) is right". More usual in Gaelic is to say "ceart gu leòr" (right enough, as in "fine"). Also, the general tendency in Gaelic (unlike, e.g. in Welsh) is to use the "he" word for an unidentified "it", rather than the "she" word.
Tha cù agam glè mhòr. 'S i Bernese Monadh Cù a th' ann.
I have a fairly big dog. She is a Bernese Mountain Dog
"fairly big" is with "dog", not separate from it, so don't split them with an "agam". Also, "glè" actually means "very" (I think it comes originally from "gu lèir" = "completely": imagine one of the modrens saying "I have a totally big dog" and you'll see how "with-it" Gaelic is...). Anyway, for "fairly" you might try "gu math" (unless that's "I have a well big dog"...), so try "Tha cù gu math mòr agam", or "Tha cù mòr gu math agam", or "Tha agam cù mòr gu math" or some such. As for the "Bernese Mountain Dog", this is some make of dog I've not come across: I wonder what the name might be in the original Swiss? (probably Swiss German in the Bern area). Maybe all dogs there are like that so they just call it "hund" or something. But anyway, if you're saying something is a Bernese Mountain variety, that bit goes after the noun. So, in English, "Bernese Mountain Dog", Gaelic order "Dog Mountain Bernese", possible tronslotion "Cù nam Beinn Bearnach", or maybe Cù Bheanntan Bhearn". "Monadh" is used for "moorland", usually at a height, and might be appropriate for the Bernese Mountains, but I'm not familiar enough with the area to say.
'S i Eowyn Anduin an t-ainm a th' air.
Her name is Eowyn Anduin.
"'S i", or just "Is". You could even use "'S e", in spite of the fact you're talking about a female dog. But what you've actually put is more like "His name is Owen Andean", as you've used "air", which is "on HIM". You need to use "oirre" (= on HER) when referring to a female object (either actually or gramatically female).
Eowyn tha i nighean anns an fiolm "The Lord of the Rings".
Eowyn is a girl in the film "The Lord of the Rings".
Now, you seem a bit troubled by "nighean" and "caileag": either will do. Yes, "caileag" is the "official" word for "girl", while "nighean" is technically "daughter", but it can also be used for "girl". So, do not fesh yourself. Now, in the English sentence, as it's written and not read, I can't tell whether you mean to stress any particular bit of it, e.g. are you saying "EOWYN is a girl in the film X, or "Eowyn is a GIRL in the film X, or "Eowyn is a girl in the film "LORD OF THE RINGS", or "Eowyn IS a girl in the film X", or Eowyn is a girl in the FILM X" - or is it just unstressed? Now, I may be wrong, but I have a feeling that you're wanting to draw attention to the NAME OF THE LASS, i.e. Eowyn. In which case, the sentence would start :-
"Is Eowyn | 'S i Eowyn | 'S e Eowyn..."
...and then "nighean" or "caileag" - no "tha". However, it's a somewhat unusual sort of Gaelic sentence. More usual would be a bit of stress on the GIRL, e.g.:-
"'S e nighean|caileag san fhilm "Laird of the Rengs" a th' ann an Eowyn"
Note "...that is in Eowyn". You could also start that "'S i" or "Is", of course. With out stress, there's the even stranger looking:-
"Tha Eowyn na nigheann/caileag san fhilm "Morair nam Fàinneachan"
or "Eowyn is in her girl in the film X"
But it's not at all right to say "Eowyn tha nighean..." etc. You just can't use "tha" (or "bha", "bheil", "robh", "bhiodh", "bidh", "bi" etc) to join two nouns together. You need to use "is" or "bu" to do that.
I am over the top with all the guidance 
Cuiridh mi gu ceart e.