1. Is it permitted to ask a gentile to take a baby on Shabbos to a kiddush?
Unless there is a great need and concern for a major loss, it would be forbidden.
Although many authorities write that our public domains do not have the status of Torah public domains, in which case asking a non-Jew to take a baby to shul would be a shevus de-shevus, doing so is only permitted for the sake of a mitzvah, or out of concern for a a great loss.
Sources: See Magen Avraham 331:5, and explanation of Mishnah Berurah, concerning bringing a baby to shul for his own bris (see also Be’er Heitev 339). Some are lenient (in particular nowadays), and permit the baby to be brought to shul by a non-Jew, but this is only true for the sake of the mitzvah of milah. We find that Beis Yisrael (no. 67) permits asking a non-Jew to bring the cholent through the public domain, but once again this is only for the mitzvah of eating hot food on Shabbos (and out of an added concern that cold food might cause health damage), and not for a regular need or desire.