Can I sell my house and keep my equity if my bankruptcy was already closed?

Of course. Your equity is, by definition, the sales price, less payment of the secured debt and less payment of sales costs. Your equity would have been protected by your exemptions declared on Schedule C of your bankruptcy.

On these facts, yes.

It sounds as though you are getting your terminology confused. A case will be DISCHARGED if it is completed correctly and the debtor is entitled to bankruptcy relief. A case will be DISMISSED if for some reason the debtor does not or cannot complete what is necessary or, if the debtor is not entitled to the relief requested. If a case is dismissed then the debtor is right back where they began without any debt relief. A case is CLOSED only after one or the other of the previous things happens, and only after all the administrative work in the court has been completed. Until your case is closed, or until your trustee abandons the asset, you do not have the ability to sell your home. If the property has equity below the amount that you can (and have) claimed as exempt you can file a motion for abandonment and ask the court for an order which will permit you to proceed with a sale.

Regardless of whether your bankruptcy was discharged and closed or dismissed, once it's over you can keep the equity in your home.

Yes, if your case was closed and the trustee abandoned the asset you can sell your house. You should contact your attorney and/or trustee to confirm this.

It is impossible to answer your question of the facts presented which do raise issues. Please contact a bankruptcy lawyer if you do not have one.

Without all the facts I cannot answer this. It depends on why your case was closed. This is a question for your lawyer.