The answer is no to all questions. To the first and second questions, an individual can never sign for someone else, even a spouse, and have it be binding unless they have a written power of attorney (POA) signed by that person specifically allowing them to do so. If that is the case, she must present the original power of attorney at the time of signing and sign her own name with a tag that says “attorney in fact for (spouse’s name) by POA” or something to that effect. You should keep a copy of the POA with the purchase and sale agreement.
To the third question, it would not be necessary for both the husband and wife to sign the purchase and sale agreement. They can both be named as buyers on it even if only one of them signs. Not so for the sellers, however. All individuals having title to the property being sold (or business if no real property is involved) must sign the purchase and sale agreement to signify their agreement to relinquish title (or all interest in the business) upon closing of the sale transaction.