Question

Especially after a rainstorm, Santa's Little Helper likes to sit in his porch swing and think. is this a fragment​

Answer 1

No

This is a complex sentence. The sentence has two parts, a dependence clause and an independent clause. The dependent clause is "Especially after a rainstorm" and the independent clause is "Santa's Little Helper likes to sit in his porch swing and think." If this sentence was only made up of a dependent clause, then it would be a fragment.