No. Independent is an adjective. Independently is the adverb form.
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It can be considered one because it connects independent clauses. But there is some question whether (as an adverb) it still requires a semi-colon rather than a comma.
It can be, when it is used to mean "freely" (e.g. the bottom half was dangling free, the fish swam free). Otherwise, free is an adjective meaning independent, unattached, or at no cost.
The word he is a pronoun; an adverb modifies a verb or an adverb.
No, it is not an adverb. The word dollar is a noun. There is no adverb form.
Yes. An adverb can modify a verb, an adjective or another adverb.