It's because numerals higher than 4 (ie. 5+ and indefinite numerals) are not viewed individually but as one group. Then you say what the group consists of and for that, you use the genitive plural. You use the numeral in the nominative (devět), the verb in the singular (je) and the noun that develops the numeral in genitive plural (míst). The pronoun "takový" is so-called "přívlastek shodný", it's attached to the noun (míst) in the sentence, so it will also be in the genitive plural.
Yes, exactly, but the pronoun "takový" stands in front of the noun and it has to agree with it (in gender, number, case). So when you talk about 3 places, you use verb in plural, you use the noun in the nominative plural (it's the subject), like you did perfectly and "takový" will also be in the same case/gender/number as the noun (nominative neuter plural).
"Jsou tři taková místa."