I'm on the fence with this one.
You have no contract - This is a red flag. There needs to be a contract, or she is not contractually obliged to follow any orders. What NI is being paid? Is tax being paid? Is this slave labour? Is she a citizen of said country, or is this why there is no contract?
You say your dog is left all day long and crying - You do not specify if you have heard your dog crying. You do not clarify if you have asked your neighbour if they heard the dog; all I've read are assumptions, so far.
You are about to let her go with no contract and no real fair dismissal policy. Do you have insurance and leagilities in place? Were you aware that paying someone without registering it with HMRC, is potentially illegal.
Having friends over is not doing any harm - If it is, then I would raise an issue regarding theft and damage. She should, however, ask. It is polite to do so.
How is she having friends over if they are out for the entire day?
How were they able to send photos of the dog if they are out all day?
Why is your neighbour snitching on the cleaner? Were they causing them issues? Is life not already hard enough without penalising someone for having company?
Does she leave your home a mess?
What are her cleaning standards like?
What will you lose, if you turn this molehill into a mountain?
Was your dog distressed when you got home?
Was the dog distressed in the photos?
How do you go anywhere at all, during your daily life, if your dog is that bad?
Have you spoken to a vet regarding their behaviour and had a discussion about medication? Dogs can in fact suffer with stress, anxiety, depression, attachment issues/disorders (especially rescue dogs) - You can requesting calming medication from your vet. You wouldn't leave them with a broken leg, so it's the same for their well-being; they need medicating.
If you have discussed medication, or already tried - Then your vet needs to try a different course of action. I am sure if it is as bad as you say, they won't leave them without help.
I'm trying to make a balanced comment on what I have read, rather than making a shot-gun decision of, "Fire the witch!"