If they established themselves in the Caribbean, they would facilitate easier links between Eastern North America, Mesoamerica, and the coast of Brazil which would be huge. And then if they got back to Africa by following the currents, they could introduce cassava to sub-Saharan Africa and eventually corn and other non-Andean (not likely to have much contact with the Andean cultural area, sadly) New World crops to the Old World. Would be cool if the Eastern Agricultural Complex could get involved somehow, and Europeans making as much or more use than the American Indians did.
Will arrival of people from the Old World cause pandemics of infectious diseases among natives just like the arrival of Spaniards? Would they be immune to European diseases later?
I doubt major diseases would be carried with the fleet. They'd burn themselves out there, if they were brought to begin with (abandon a ship with the diseased to its fate). But possibly some diseases might be transmitted through asymptomatic carriers, but they'd cause the deaths of maybe 5% of natives at most.
1 - Diseases area always evolving. People from 1300 were not prepared for the Black Death, people from previous times would not be prepared too.
2 - If your refugees bring some disease with them, that disease will burn out and disappear quickly, without the disease around the chance of the population keep resistance against it for thousands of years is close to zero.
3 - Even if you had brought many domestic animals to the new world and the indians started to use them, it would not give the indians resistence to old world diseases. What it could happen, maybe, would be that the New World could give birth to a new disease strong enough to spread to Europe and Asia like the Black Death, maybe.
Yep. But a disease introduced to a high-population area like the Andes or Mesoamerica would survive and become endemic, wouldn't it? The evolved form would cause a nasty plague in Europeans, although it would be like a bad smallpox epidemic and not like the Black Death or smallpox was to the American Indians. The good part about Carthaginians is they might be able to transmit some agricultural knowhow, along with metal plows and such, which might be beneficial to the people of Mesoamerica, the Woodland cultures of the Eastern Agricultural Complex, etc. Ideally that and the cultural change they'd bring about would mean a higher population density in Eastern North America and thus an environment for disease to be more established in that it gives immunity.
If Carthaginians introduced influenza to the Americas (which would kill a fuckton of people as it did OTL), then American influenza would evolve in isolation to Eurasian influenza for centuries. When reintroduced to Europeans, it would create a very bad influenza plague, about as bad as the Spanish Flu if not worse. But I don't believe you'd get a new Black Death out of it.