In order to have standing in a lawsuit you must be the party who holds the note. If the note has been sold to another party then you have the right to make a motion to challenge standing and have the case dismissed if the lender who began the litigation is no longer the owner of the note.
The original lender can assign its rights under the loan documents and/or can assign its rights to proceed in the lawsuit to another party. At least in Florida, the new party may continue the action in the name of the original lender. If the new party does not have possession of the original note, then it may have a problem getting a final judgment of foreclosure, since the original note has been held by some courts to be necessary to foreclose. You can challenge standing of the new party to continue the action in any event, and they will need to show a written assignment of the loan documents and/or possession of the original note.
Actually, the loan was probably not sold. Most home loans are owned by a securitized trust that sells shares to investors. What appears to you as a new lender is simply a different servicer. Home loans are serviced by businesses that collect each payment and forward it to the trust. When a loan goes into default, it is often transferred to a servicer that specializes in default servicing.
Unfortunately, the loan can still be foreclosed. In fact, the transfer is often a first step in foreclosure.
Yes, they can and they do. The sale does not change anything for you if it is done properly - it only means that there is a different decisionmaker, but the contract you made when the loan was originated stays the same. And, for the most part, those notices that you receive relate to a transfer of servicing, not a transfer of ownership. The servicer is the "front end" the entity that takes your calls, sends you bills, takes your payments. It is normally a different entity than the owner of the loan.
You should look for an experienced foreclosure defense attorney to help you. While servicing can be transferred and loans can be sold without it impacting the foreclosure case, it is often true that mistakes are made which may provide defense opportunities.