When I first embark the ducks lay fertile eggs but not later

Normal egg-layers lay eggs roughly once per season.

However, and this is the point you're having problems with:

War animals never lay eggs; or, if they are not egglayers, never breed.

I've never had occasion to find out if animals find a nest box while on a chain, but I suppose they should.

But again; captured war animals will not breed.

Ever.

The dwarves gravitate to the wagon because it is a fall back meeting area. If you make any other meeting areas then they will congregate at those instead of the wagon. They will still split themselves among the available meeting areas other than the wagon. If you deconstruct the wagon then they will wander all over the map unless there is another active meeting area.

There are several ways to make a meeting area. One is using Zones. You create a zone somewhere inside your fortress and enable it as a meeting area. Your dwarves will then congregate there when they have nothing better to do.

Certain rooms defined from furniture also function as meeting halls. The most common such room is the Dining hall.

If you use meeting areas you should be sure to avoid generating jobs that need to be done outside, such as fetch water (if you have no interior water source), hunt, fish, or gather items among others. Also note that any animal with the grazer property will starve to death indoors unless you have cave fungus.

As for burrows, it sounds like you have the alert portion figured out quite well. However burrows are not points but volumes. You need a much larger burrow if you want to accomplish anything. Your dwarves only stand around in those few spaces because that is the entire area of the burrow. For it to work properly the burrow must include every square that the dwarves might have a task in, including un-mined tiles that you want to dig out.

You do not need an alert to restrict a dwarf to a burrow. You can add dwarves to a burrow and that will restrict them to that burrow.

Note that burrows only stop tasks from occurring outside the designated area or using resources from outside the area. Wandering dwarves and animals, specifically baby and child dwarves, will still leave them on occasion. Civilian alerts will provide more restriction of dwarven activities, but I have still had very young dwarves wander out of my burrows during an alert.

The meeting area information should be enough to keep your dwarves inside most of the time, but still allow them to go outside when something comes up.

Hope this helps.