It could go either way
Let's dispose of the two trivial cases:
- if all parties agree there is a valid lease, then there is a valid lease;
- if all parties agree there is no valid lease, then there is no valid lease.
Now we get into the weeds, and it all depends on the various parties' circumstances and positions. There are four people involved: two possible tenants, Rachel and Jared, and two possible landlords, Robert and Elizabeth, and each could be seeking to have the contract affirmed or set aside and mounting various arguments to that effect. To keep this from getting out of hand, I'll confine this answer to just the questions you asked - there is enough material in the scenario for a term paper on contract, property and agency law.
Would it matter whether the landlord's spouse is on the title of the house or not? If the spouse is on the title she may be able to enter into a contract agreement in her own right. On the other hand, the lease may still be void because it contains conflicting information as regards to who is actually entering the agreement.
Absolutely. It also matters if you mean on the title "as well" or on the title "instead" of Robert.
If Elizabeth is the sole owner, she acts on her own behalf, and the lease is valid. The fact that it has Robert's name printed on it would probably be treated as an irrelevant clerical error.
If Robert and Elizabeth are joint owners, then any contract must, of necessity, have both of them as one of the parties. Elizabeth can act on her own behalf, but she must be acting as an agent of Robert for the contract to be binding. Again, the fact that only one of them is named in the contract would probably be treated as an irrelevant clerical error.
Similarly, if Robert is the sole owner of the property, then Elizabeth must be acting as his agent to enter the lease.
Would it matter if the husband/wife could prove that he authorized her to sign on his behalf? Would signing her own name instead of his name be problematic?
Whether Elizabeth is or is not Robert's agent is a matter of fact - she either is and the contract is binding, or she is not and the contract is void (q.v. apparent authority below) - see Jarjo v Patterson  NSWSC 1049. So, if Robert authorised Elizabeth then she is his agent and the contract is valid whether she signs it Elizabeth, Robert, or Micky Mouse.
What if the other party does not dispute that she signed on her husband's behalf?
It doesn't matter whether Rachel and Jared know the true situation or not with respect to the validity of the contract. If there is deceit or negligent misstatement by Elizabeth, this may give Rachel and Jared grounds to sue Elizabeth for damages caused by this, but the contract would still be void.
However, if it was reasonable in the circumstances for Rachel and Jared to believe that Elizabeth was Robert's agent, then they might be able to claim that she was acting with apparent authority even in the absence of actual authority. From the question, we simply don't have enough information on the circumstances to decide if this argument has legs.