Rosh ha-memshala mekhake la-bkhirot ha-ba’ot.
No, בָּאוֹת coming is an attribute (originally a participle, here used like an adjective), therefore בְּחִירוֹת needs an article too. A construct would mean something like the election of the coming (women).
is literally head of the government. In Israel this is the prime minister but it varies from country to country so would the meaning of the word change depending on the political system?
Well, it depends. For example אנגלה מרקל is certainly our רֹאשׁ מֶמְשָׁלָה, although Angela Merkel's German title קַנְצְלֶרִית chancellor is usully the one used in Hebrew too.