If airport weather condition is below minimum, but pilot report that flight weather condition is above minimum, Does controller can issue approach clearance to pilot?
If weather condition get worse than published minimum, should pilot cancel his approach and execute missed approach even in final approach phase?
You haven't told us what country you are referring to. I am going to talk about the United States regulations.
There are two separate regulations that come into play. Part 91 and Part 121 or 135.
Under Part 91. The pilots are allowed to commence any approach (irrespective of weather) and determine if they can continue to land based on the criteria listed in §91.175(c).
Under Part 121 / 135. The pilots can only commence an approach (proceed past the final approach fix) if the latest weather received is above landing minimums. Mainly, pilots are concerned with visibility. They will not be able to use "flight visibility" as the weather visibility is most likely different as the airplane descends towards the ground. The pilots must use the weather reported from the airport.
To answer the second question, if the pilots are past the final approach fix and new weather arrives indicating the airport is below minimums, the pilots are allowed to continue the approach to minimums to see if they can land per §91.175.
There are many places where ATC doesn't give clearance at all. A VFR flight can be completed without any ATC communications of any kind.
Assuming an ATC clearance is required, ATC doesn't give an approach clearance based on reported weather, they give a clearance and it is up to the pilot to fly the approach and execute a go around according to the approach procedure if visual contact with the runway or approach lights isn't established. Often weather conditions are changeable, what was reported in the last METAR can be significantly better or worse just a few minutes later, so a pilot can give the approach a try.
With very few exceptions a pilot can go around at any time, even on a late final. From a weather point of view it is a judgment call to be made by the pilot, if visual contact with the runway is lost for example. Notable exceptions I can think of are airfields like Aspen, where a go-around is impossible after a certain point in the approach.