"I don’t know if my DF’s will went to Probate. Should it have?"
From what you have described it seems as though it should have.
"...she told me that nothing could be done “behind my back” because there is a will and I’m an Executor."
That is correct, nothing can be done behind your back lawfully.
However, that doesn't stop things happening unlawfully.
For example, I'm aware of a situation in my extended family where the parents owned the house as tenants in common with each half being left to the children and the surviving spouse having a life interest.
Both the children were named as executors but they did not (even more so since) speak to each other. When the father died, one of the children forged the signature of the other child on the form you need to complete for probate indicating that they did not wish to be an executor.
So the one child managed to get probate while totally sidelining the other child. All sorts of stuff hit the fan when the other child found out about this.
If they were tenants in common then there will need to be probate and the will becomes a public document.
You can search for your father's will here, it will cost you £1.50 to download it:
https://probatesearch.service.gov.uk/help
If you were named as an executor and you were not involved in gaining probate then it may be that something dodgy has been going on.
"I don’t know who the property is currently registered to at the Land Registry"
This is a simple matter to find out, it costs £7 to download the information. You want to download the Title Register for the property.
https://www.gov.uk/get-information-about-property-and-land/search-the-register