Well, "Frantishek" is certainly a big no-no. "Frantisek", on the other hand, is possible and accepted.
Languages that use the Roman script write foreign names as they are - without any phonetical transcription hijinks. Or, it's always possible to remove the foreign diacritics and use plain Latin characters, if someone's feeling lazy.
Take, for example, the French playwright Molière - his name is written like that, with è in both Czech and English, even though neither language has that character. It's possible to replace it with a plain "e". Between Roman script using languages, letters are never replaced by other letters (or diagraphs) though.
It's a different approach than, for example, Russian has when transcribing English names or, vice versa, writing Russian names in English.
But if František moved to the U.S. for example and dropped the diacritic, then his name would be pronounced with a "s," which is not his name. A good example of this is, I know Hungarians in America whose names have "sz." They delete the Z in English so that it's pronounced correctly. Or on the other hand, American Poles leave the Polish spelling and allow for their names to be mispronounced in English. Frantishek is at least pronounced correctly
Yes, that's true, that's how it's handled when you want to be assimilated (or domesticated? :D). The same way, a French guy called Jacques could choose to change his name to "Žak" if he wanted to live in the Czech Republic for the rest of his life and pose as a Czech. But that's a rather extreme situation, changing your name - in most other circumstances, he would simply keep his name Jacques and told people how it's pronounced. If František doesn't want to become a full-fledged American and pledge allegiance to the flag, he will remain František even in English. And in case he does get the green card and whatnot, he will probably opt for "Frank" anyway.