I'm going to answer your questions in reverse order.
Is there a specific distance threshold that you cannot cross above the approach speed?
The short and simple answer is yes, there is such a point. More precisely there is a point, such that if it is crossed above the approach speed (with some margins) a missed approach becomes mandatory. The missed approach is not necessarily performed exactly at that point, but a landing is no longer permitted.
In this day and age an airliner is nearly never flown in such a manner that there is a gradual deceleration all the way to landing. Where that point, known as the stabilization point, exactly is depends on the aircraft flown, the airline policies and what type of approach is flown.
What has forced the industry and regulators into defining such a point is the finding that many accidents have been caused by what is known as an "unstabilized approach": Wikipedia: Unstabilized Approach.
In really broad terms a few different ways to define this point:
- 1000 ft AGL in IMC (sometimes the same in VMC also)
- 500 ft AGL in VMC (sometimes 1000 ft in VMC also)
- The Final Approach Fix (in non-decelerated non-precision approaches)
- Some much lower altitude for circling approaches and visual circuits
These will vary depending on the airline and the regulatory authority, so there is no definite reference where you can find these from. There are also other criteria than the speed that must be fulfilled at the stabilization point. There are some sources you can refer to for examples:
Once the initial approach fix is reached, what are the different "speed stages" to the runway?
Referring to the first section of my answer the initial approach fix is not critical in determining your speed profile. Rather you determine the point at which you must be stabilized at approach speed first and work back from there and in this way determine the point where you would first have to start slowing down and configuring the aircraft step by step (flaps, slats, landing gear).
In an ideal world you would want a continuous deceleration from the point you decide to start decelerating all the way until your stabilization point. Flaps would be extended as soon as the max allowed speed for the next configuration is reached and the landing gear in a sequence that usually comes with some recommendations from the aircraft manufacturer.
That means that the location of the point where you start your initial deceleration depends on many variables: aircraft type, aircraft weight, altitude, wind conditions etc. In addition to that there are factors that make the world less ideal, like air traffic control, procedure speed limits etc.