What is the average speed used by modern jet fighters when in dogfight?
Modern Air Combat Maneuvering takes place between about 350 to 600 KIAS; There is no specific speed but each aircraft has a specific optimal cornering velocity which gives it the greatest rate of turn in order to maximize its maneuverability.  Pilots will seek to maintain this speed when entering a merge as it can give them the greatest maneuvering advantage in order to turn inside on opponent or for jinking to evade a cannon or a missile shot.  As an example, the Lockheed Martin Block 50/52 F-16C has a cornering velocity of 420 KIAS.

Traditionally air combat has been about energy management.  Fighter pilots have something in common with economists in that they are engaged in the art of managing the flow of currency; in the case of the pilot that currency is energy and not money.  The total energy state of a fighter is:

current energy state = energy in - energy out.  

The current energy state is the sum of the kinetic energy, proportional to the square of the velocity of the jet and the potential energy of the jet, which is proportional to its height above the ground.  Energy leaves the system in the form of parasite and induced drag acting over a distance as well as loss of altitude, which is proportional to the flat plate area of the jet, load factor, AoA, etc.  Energy enters the system at a rate proportional to the amount of jet fuel burned; this supply is limited to the fuel onboard which again affects the mass of the jet, the induced drag, etc.

So keeping this in mind, a good fighter pilot will seek a way in which he/she can maneuver through space, gaining the best speed and turning performance for the least amount of energy consumed, in order to land in an advantageous position to employ the weapons against an enemy jet.  These kinds of speeds vary between if the jet is on the offensive or defensive in relation to a threat.

So a pilot maintaining that optimal energy state is in the best position to attack, or defend against an attack, and a pilot in a very low energy state is very vulnerable - "low and slow, you're outta dough" - and could be quickly killed in a multi-jet furball. 

The cornering velocity is preferred for this reason if maneuvering is anticipated (a given if the jet is about to enter a merge).

With the advent of terminal homing, beyond visual range missiles allowing pilots to stand off from the fight and shoot a threat before it could get close enough to turn with them, higher speed cruise, along with a capability to maneuver well at those higher speeds has become an advantage.  It allows a pilot to more quickly close with an enemy aircraft into the missile's 'no escape' zone, fire, and then turn and dash from the threat, minimizing their presence in an enemy's 'no escape' zone whereby enemy missiles simple don't have the available energy to catch you.  This was the basis for supercruise - supersonic cruise without the use of afterbruners - on the ATF / F-22 program.  Combined with good frontal stealth to minimize the detection range by enemy fighters, supercruise could give a Raptor jock a major advantage in the BVR areana - near invincibility in BVR engagements against non stealthy fighters, if the published results of the Red Flag and other military exercises are to be believed.