On the structure of "search for weapons and bands of pro-Hussein fighters still holding out"

Solution 1:

To hold out here means ‘to maintain resistance, remain unsubdued; to continue, endure, persist, last.’ Holding out is the non-finite -ing form of the verb. A paraphrase would be ‘pro-Hussein fighters who continue to offer resistance.’

Solution 2:

The boldfaced part is a noun phrase with a reduced relative clause. It's another case of Whiz-deletion. The original was something like

• ... bands of pro-Hussein fighters who/which/that are still holding out.

where Whiz-deletion, as is its wont, deleted the boldfaced part -- the Wh-word subject of the relative clause, and the auxiliary be of the progressive construction that follows it. This is very normal behavior for English.

So, in order, the questions:

1. Holding out is not a gerund; rather, it's what's left of the progressive construction after are got deleted, namely the present active participle (the -ing form) of the intransitive phrasal verb hold out.
2. The subject of (are) holding out is who/which/that, which got deleted. This relative pronoun, however, is coreferential to its antecedent, pro-Hussein fighters, so it means the same. But the (deleted) relative pronoun is the real subject of the relative clause.
3. This sentence is not different from a full, untransformed relative clause. It's just that English often deletes predictable syntactic markers to make things shorter; they mean the same and they work the same. Generally the full clauses are considered somewhat more formal, but not always.