IIRC air to air missiles use a "boost-sustain grain" motor that involves the missile's motor boosting quite violently for a relatively short amount of time to get to top speed and then switches over to a slow-burn mode where the motor burns a lot more slowly and with less thrust but over a much larger interval of time.
The "All-Boost" missile motor you were initially thinking of was tried way back in the days of developing the Falcon series of missiles. For that particular missile when the "All-Boost" motor type was used the missile accelerated all the way to high-supersonic speeds but the burn time was so small that when the motor burnt out the missile was forced to glide all the way to the target. This wasn't desired because the air force needed a missile for their interceptors that could climb up to high altitude above the launch platform to engage enemy bombers in a "look-up" head-on engagement. Only a sustainer motor could efficiently provide a missile with that flight profile. That, and an all boost motor would cause excessive drag and aerodynamic heating to the point where the effective range of the weapon was severely limited once the motor burnt out leaving the thing gliding at Mach 3 in the face of such large drag penalties associated with a body moving that fast un-powered.
A sustain motor simply burns for a long time and provides enough thrust to counter drag and keep the missile at a certain cruise speed. However, by cruising efficiently the missile can slowly climb up to gain potential energy so that it has the altitude to engage targets above the launch vehicle and/or engage targets at lower altitude and much much further away by converting potential energy to kinetic energy for maneuvering against targets at extreme ranges. Such a motor is best suited for head on engagements at extreme ranges against enemy bombers and the like. The problem is the sustain-motor doesn't boost the missile to high speeds for a quick high-speed intercept. Its top speed was too heavily dependent on the launch-platform's speed at launch.
The boost-sustain motor works by combining the best of both worlds. The motor is specifically shaped to burn brilliantly for a short amount of time to get the missile to supersonic speeds that are typically far above the launch platform's speed at launch. Then the motor slows down after a few seconds to a longer-burning cruise/sustain burn profile to maintain speed and even perform a climb for look-up engagements and to increase effective range as planned for the aborted AAM-N-10 Eagle and utilized in the AIM-54 Phoenix and later AMRAAM variants.
The Falcon family also adopted the boost-sustain grain motor type too.
For your second question, there's the AIM-152 missile design submitted by GD/Westinghouse.
http://www.secretprojects.co.uk/forum/index.php/topic,2548.0.html
It does exactly what you specified. It boosted to a high speed, dropped the booster motor, then switched to an internal motor to continue flying at high speed and then to get up to a lofted altitude. The missile could then shut down its motor and then restart it again in the endgame to surprise the enemy and even use a thrust-vectoring nozzle to boost its way downward (to convert PE to KE) while performing skid-to-turn maneuvers to outmaneuver the target at ranges of around 100nm.
There's been some speculation that the JDRADM will use that flight profile as well. Multi-pulse rocket motors are what they are referred to. Boeing's and LMT's JDRADM concepts both might use a regular boost-sustain pulse to both boost and sustain their velocities only to shut off and use another pulse to outmaneuver the enemy in the endgame.