I'm learning Czech and in my book it says on last names of women end in -ova, and if your name ends in ek, ec,... that you remove the e and replace it with the-ova. So if thats so would Carter be Cartrova? if not what would it be?
Well, for Carter (as an example), it would be Carterová, as the surname is not of Czech origin. But for Sládek, the female surname will be Sládková, Hošek - Hošková etc. Interesting is the surname Fiala, as the female form Fialová can also be derived from Fialový. Surnames ending by -ý (adjectives) adopt -á for females, identically to regular adjectives; e.g. Holý - Holá, Nový - Nová etc.
Yes; Tomášek - Tomášková, Mašek - Mašková, Hašek - Hašková, Tomeček - Tomečková etc. etc. There are hundreds of Czech surnames ending -ek, as it is a suffix used in diminutives.
My name is Paige Ryan. While in the CR should I change it to Paige Ryanova? Or is there any other way to make my name look more czech?
Well, in everyday contact with people you may introduce yourself as Paige Ryanová, but all your official documents should keep your original name. I am sorry, but I do not know about any other way you could make your name sound more Czech, unless you get (and I am sure you would, if you stayed in CR long enough) some Czech nickname or "Czechized" version of your first name, like Pažinka, Pažulka, even Pažitka (chives).