Lawsuits where you may recover money are “assets” in either Chapter 7 or in Chapter 13. Thus, you have to disclose the suit in your paperwork (including class actions). In Chapter 7, the Trustee may keep all or part of a recovery, and may prosecute a suit as he sees fit. The amount that you can keep will depend on the amount that you can “exempt” (see, question 3, above). You should discuss this kind of “asset” with the lawyer before filing because hiding assets can be “bad news” for all involved.
In Chapter 13, the lawyer in your non-bankruptcy suit has to be approved by the court, and you may have to pay part or all of any recovery into the case to pay creditors. That question is resolved after the suit is settled by a motion filed with the court. You have to discuss your specific situation with your lawyer to get a solid understanding of how you may be affected.