To determine the number of basic land types among lands you control, look at the lands you have on the battlefield and ask yourself whether the subtypes Plains, Island, Swamp, Mountain, and Forest appear within that group. The number of times you say yes (topping out at five) tells you how powerful your domain abilities will be.

How many lands you control of a particular basic land type is irrelevant to a domain ability, as long as that number is greater than zero. As far as domain is concerned, ten Forests is the same as one Forest.

A number of nonbasic lands have basic land types. Domain abilities don't count the number of lands you control -- they count the number of basic land types among lands you control, even if that means checking the same land twice. For example, if you control a Tundra, an Overgrown Tomb, and a Madblind Mountain, you'll have a Plains, Island, Swamp, Mountain, and Forest among the lands you control. Your domain abilities will be maxed out.

You do what the spell says in order, so you'll put a new basic land card onto the battlefield before you determine the value of X.

Exploding Borders will still deal damage even if you don't put a land card onto the battlefield.

Exploding Borders has only one target: the player. If that player becomes an illegal target by the time Exploding Borders would resolve, the entire spell is countered. You won't get to search for a basic land card.