What kind of heat treatment are you planning to do?

Induced phase transformation depends on temperature and time. You can repeat simple quenching heat treatment numerous time if sample is austenitized before hand.

Everything depend on what heat treatment you are planning to do as David Bombać wrote. Look on the phase transformation curves and You will get the answer how the treatment affected the material.

yes you can, there is no rules for the heat treatment you can do whatever you want, to obtain the material properties that serves your need. However, as you are treating steel tools, you should not increase the temperature during the heat treatment close to the material melting point also the final product should have some degree of brittleness. You also mentioned to the material strength, sure, the material strength, ductility, Brittleness,  ability to absorb shocks all these properties will change if you reheat the material.  Generally speaking materials cooled or solidified at slower rates will have relatively large and relaxed grains (more like ductile material); whereas materials cooled and solidified at higher rates will have smaller grain sizes, and will have less ability to absorb shocks (easy to broken), the material properties will be more like brittle material properties.