Its important that you don't think of specific counters to specific attacks .
There are so many ways that one could potentially be attacked that its impossible to have a specific counter to each one.
If your head is filled with endless lists of counters it is debatable as to whether you will be able to respond spontaneously to attack.
We can make this easier by thinking more in terms of types of force and the direction that the force is going in.
Is the force coming straight in at the centreline , is it circular , is it coming from above , is it coming from below, or from the side.
In answer to your question for an axe kick , we should be in the first instance trying to jam the kick and follow up before it even gets off the floor.
But if we are only able to pick it at a late phase , then because the attack is coming from above we can use a Seung Bong (double bong sau) that is both arms raised high above head in parallel formation , close together for maximum resistance , we then redirect the force down and to the side while pivoting , then you can follow up with a Fak sau.
Alternatively you could use the same technique but team it up with a side stamping kick to the support leg.
I have also just charged in with a Dai Sau and punch before , the Dai Sau is just raising your arm from the guard straight up like one side of a triangle , this has the effect of jamming their leg up high and they are knocked off balance .
We do give chase and we use various combinations of low chain kicks which enable us to cover ground rapidly and with some fire power.
The reason is we don't want that person in front of us to be in his comfortable kicking range , we want to get in close where we are comfortable and close him down with our hands.
A good exercise for being able to stop someone getting away from you and being to react effectively when they do , is this one .
When you are doing Chi Sau have your partner break off suddenly at random times , in the first drill we will try to feel for when he is about to break off and try to latch him back in with our hands .
The next drill he has got away from our hands and so we will use a thrust kick or medium heel kick as he is retreating .
The last drill he will break away and retreat as fast as he can for several steps , in this one you will use a sequence of kicks to chase him as soon as he breaks contact then close in with hands .