Are there much in the way of hiking?

Is there hiking trails to the mountains

Yes trails go around the lake and out into the woods. They are nice trails with varying grades.

Yes, but starting from October the park is getting close. Therefore you have to park outside and hike yourself

yes

Yes they do

There are several trails, between 1/4 and 3/4 mile in the park. http://www.pulaskigatewood.com/pulaskivirginiaparks.php There are also trails outside the park (on the road leading up to the park).