Carthage was a Tyrian colony, not a Phoenician colony. While Tyre was a Phoenician city, we need to understand that the cities of the Phoenician coast as independent cities as the Greek cities were.

Now, we do have some evidence of contact between Tyre and Carthage. When Alexander was conquering the Levantine coast, he laid siege to Tyre, which had remained independent of Persia. When Alexander laid siege to the town, the city was about to begin a religious festival and the Carthaginians had sent a delegation to participate (See Arrian 2.24). Carthage also promised to send a fleet and house the elderly and women of Tyre. Here is a link to Arrian, the information about support can be found in Diodorus and Curtius, the reference can be found in a footnote in 2.24 https://archive.org/stream/cu31924026460752#page/n153/mode/2up.

The relative veracity of all of these claims can be debated here. Frankly, I'm inclined to trust Curtius more than Arrian and I would be happy to debate this point. Regardless of the truth of these claims, I think that we can assume that there was some sort of connection between Carthage and her mother city that has some sort of religious aspect.

There is a myth of Carthage's foundation in the Aeneid that claims that Dido fled Tyre. The wikipedia article for Dido actually has some good references if you want to look them up and nobody else gives a detailed answer to your "Bonus question."