The saying is that "the will speaks from death". It follows that the executor is appointed at the time of death. Obviously, if the will has not been located, there may be a period when the executor is unaware of his legal capacity. It might happen that the nearest kin of the deceased take it upon themselves to deal with things - the obvious matter being to arrange a funeral. It would, in my opinion, be churlish of an executor to make an issue of such things unless something egregious had been attempted.
In a sense, the executor acts provisionally until such time as probate has been granted.
If (a) the estate is small, (b) all the assets can be marshalled without a Grant of Probate, and (c) the beneficiaries are agreeable to the arrangement, then the estate might be finalised without applying for probate.
If you need to ask the last question in your post, I would be concerned that perhaps not all the beneficiaries are agreed.