An accepted offer is not a sale. Until conditions are removed you have an agreement that one side can still walk away from.
A pre-approval is not an approval. It tells you a lender has reviewed the buyer in principle; it does not confirm that this lender will lend against this property at this price.
Financing conditions fail for reasons that have nothing to do with the buyer's income: an appraisal coming in below the purchase price, a lender declining a property type or a home with an unpermitted suite, or the buyer's circumstances changing between pre-approval and closing.
Which is why the strength of a financing condition is worth asking about. Which lender, how far along, and has the buyer actually submitted the property to them?
Generally yes, and it is normal practice. Continued showings preserve momentum if the deal fails, and can produce a backup offer. Some buyers dislike it; that is a negotiating point rather than a rule.
What you cannot do is treat the accepted offer as if it does not exist. You have a binding agreement subject to conditions, and how you can deal with other interest depends on what the contract says. That is a conversation with your REALTOR® about your specific contract, not a general rule.
Yes, backup offers are permitted and can be a sensible protection. A properly drafted backup sits behind the accepted offer and only comes into effect if the first collapses, which means no gap where your home returns to market cold.
The drafting matters — a backup offer that is not clearly conditional on the first contract ending can create serious problems, including the possibility of being bound to two contracts. This is one to have written carefully and reviewed.
If conditions are not satisfied or waived by their deadline, the contract typically comes to an end and the deposit is ordinarily returned to the buyer. The specifics depend on how the contract was written, so confirm your position rather than assuming.
Practically, your home returns to the market with days on market already accumulated, which is one more argument for scrutinising the strength of an offer at acceptance rather than only its number.
The offer most likely to close is not always the one with the highest number on it. Two weeks lost to a deal that was never going to complete costs more than the difference between the two offers you were choosing from.