No. It will not.

Yes. All of your provinces -- including both core and captured provinces -- will receive an instant +10% morale boost. This can be critical in stabilizing newly captured provinces and preventing potential rebellions among your newly captured provinces and others that have low morale for whatever reason. Please note that the morale of a given province directly impacts resource production in that province.

Also, when you capture an enemy capital, you will receive a significant portion of the enemy's treasury balance -- I believe it's about 50% of the enemy's cash on hand. Based on personal experience, I have received widely varying cash windfalls for capturing an enemy capital, from a few thousand dollars to over a million in one instance.

In order to capture and occupy an enemy province you have to occupy the provincial center. In order to pass through a province in most cases, you must also pass through the provincial center, thereby capturing it or engaging in battle with enemy troops located there. There are a few provinces where there are alternate pathways that permit you to bypass the provincial center, but that's not typical.

Basically your units will fight ANY enemy in their way, they will either win, capture the province, and continue to their target, or die trying (they are very brave and heroic XD).

Also keep in mind that it isn't usually a very good idea to set them in a long path through enemy territory right to the capital unless you have an idea about the type of troops they will be facing, you might be sending them to death without gaining any advantage, the nature of each province (Forest, Urban, etc) should also be considered, it can greatly affect the results of any battle and also the speed of your troops.