That's almost correct, i think. More frequent is "Frantík", because it is usually used as euphemism for baby boy's genital. I think it comes from historical spa city in Czech republic called Františkovy Lázně. There is a story or legend about local "statue of František". It is said that any woman who touches the statue of the František will soon be in the family way. (The statue, by the way, represents nude young boy sitting on a sphere and holding a fish in his hand) Guess which part of his body is shining most? Yes, it is his "Frantík". :-) Final note: These euphemisms are pretty vernacular, so it's perfectly possible that people in nearby region never heard this "story about frantík's origin".
Well it actually could as Great Brittain is also translated as Velká Británie. There is also another word for great in czech "Velkolepý" in the meaning of importance and intense influence and "Výborný" in the meaning of astonishing quallity. It could also be tall František. While tall is something different than big and even czech have word for tall different from "velký", which is "vysoký" it is also often used when talking about something which is tall or big overall and tall also.
Not normally, but a person's name doesn't change just because the language changed. James doesn't become Jacques in French and vice versa.
If you install a Czech keyboard or learn the alt codes (which you'll want to do for writing in Czech anyway) it won't be hard to write them. :)