You list two distinct types of injuries which require different kinds of avoidance (which I think is the key: avoiding the injuries to begin with, yes?)
In one, you mention the kicking pads. We use a variety of pads, one is called "elephant ears". These are usually held in one hand, so, you hold two. For kicking drills, I get clever about holding one pad with one hand, and with the other target, it covers my holding hand. This way, if the kicker misses the pad, he kicks the protecting pad.
As to avoid toe injuries, there are several things to do. When kicking (and turning), always lift up. This reduces the tendency to grind the foot into the floor. Too great of a lift, and you end up doing a jumping turning kick.
Puzzle mat floors are a particular danger, because when the puzzle pieces don't align right, they can pop up, or create holes. These popups or holes can grab a toe, and if you aren't careful, can break the toe. Nothing you can do here, except lift your feet when walking on these mats.
Another injury I see in classes is during breaking routines: the board holders are always at risk. They make these finger protectors just for this purpose: when a kicker misses the middle of the board, and opts to take out your fingers instead, they end up kicking the rubber finger protectors instead.
There are other kinds of injuries, such as stretching or when being kicked. These muscular problems can be resolved with pain killer meds, or, cremes like Ben-Gay, and the like.
Other muscular injuries are due to overexertion. These are caused by lactic acid build up. The way to get around these to perform quad- and calf-stretches, which squeeze the lactic acid build up out of the muscle tissue.
And if you are looking for pain medications, there are a few. NSAID like Ibuprophin can be helpful to reduce pain caused by swelling, because, they're anti-inflammatories.
You can also use products like Ben-Gay and Tiger Balm, they have high amounts of eucalyptus, camphor, and other topical anesthetics. These can help allevite muscle damage due to stretching or blunt force.
Kicking pads, I forgot to mention about others.
So, large thick pads. I tend not to use the straps at the back for strong kickers. For them, hold the pad close to my body, then absorb the kick by moving back in the direction of the kick. This way, I'm not slapped with the pad, and, I'm moving with the kicker. Remember, strong kicks are what kicking bags are for; these pads are large to protect you - but kickers assume they're large and meant for power. Nope. But they still kick hard, so, you want to minimize impact to you. (And for younger children, you want to hold the pads by the straps and then not held to your body. This is less about protecting yourself and more about being able to maneuver the pad so that you can watch the kick to provide feedback.)
For elephant ears and other one-handed targets, be sure never to hold them with a locked elbow, or you'll inherit some severe joint damage. Hold the targets so that the face allows the kicker to properly kick with the heel, ball of toe, or instep - whatever is appropriate for the kick, and the kicker's reach and capability. It's not always necessary to be so rigid; the more rigid you are, the more you absorb the kick into you.
If you are holding a pad for a kicker who's too fast or hard for you to hold properly, you shouldn't be paired with that person. The instructor should prevent that, lest that creates a safety hazard to you.
For elephant ears, it's common to hold the target for axe kicks in such a way that is reminiscent of holding a dagger for a downward stab. In this way, you hold the target high with your fist grasping the handle and with your elbow behind the pad (but not resting against the pad). In this way, when the kicker comes down on the pad, it will slap back safely against your elbow, not your face. If they miss the target completely, no part of your body is exposed outside of the sides of the target.
When holding an elephant ear target for roundhouses, if you hold it so that the target as extended from your arm, then, if the kicker kicks too close, your hand will catch the kick and hurt like hell. So, hold it like a knife where it extends 90 degrees upward or downward from your arm. An over kick will not contact your arm or hand. However, in this hold, a kick that is too high (or too low) could still contact your hand. So in this regard, as I mentioned before, I use another target to cover my holding hand, so that a kick that is too high or too low will be protected by the protecting target instead.