1) A Health & Welfare Fund ("Fund") obtains judgment against defendant. May the Fund pursue attorney fees incurred in the collection process (tracking down defendant, attorney time in drafting writs of execution, etc.) after judgment has been entered? In what venue (Federal or State court) must the Fund pursue those fees?

2) If the suit concerned delinquent payments, may a collection action be pursued in state court?

3) If the suit concerned non-payment, may a collection action be pursued in state court?

When the suit was originally filed, it should have demanded costs. It's too late now.
You don't state whether the judgment obtained was in Federal court, state court, small claims court or what. The questions about whether state court is appropriate would vary widely from state to state depending on state-recognized causes of action and procedural rules.
These seem like questions for the attorney who obtained the judgment.