Yes, you do need to clear in and out of each country as you go, though not each island. So you will need to check out of Carriacou (part of Grenada) and into St. Vincent - probably at union island. Then back out and in again. Offices vary, but I would plan 1.5 hours +/-. At Union, you check in and out at the airport, which is about a 30 minute walk from the harbor, so that could be a little bit longer unless you get a ride.

In most cases, only the captain checks in (with all passports) for the crew (check each county's requirements). It really helps to 1) have all your papers in order and properly filled out in advance whenever possible, 2) be patient, getting pushy and impatient will not help, 3) have cash for the fees. Overall, it's just another part of the experience.

TC42 - the passages can be rougher, I would take kids if they had prior experience and I would want a flexible schedule to be able to pick a weather window. It really depends on individual preferences, experience and the boat.