Can my opponent kick my blade away, block with their other hand, or bump into me? What's actually illegal on the strip?
No, kicking your blade away is not legal: USA Fencing and FIE rules say that with all three weapons, defensive actions must be made exclusively with the weapon. Beyond that, the list of real offenses is shorter than many fencers think, and it boils down to a few ideas. Defend only with your weapon, do not hide or swap valid target, do not use contact or the edges of the strip to escape a hit, and do not turn your back. The cards answer explains how the colors escalate; here is what earns them.
Using your non-weapon hand or arm to attack or defend, such as pushing the blade aside or blocking, is a red card, and any touch you scored is annulled. A kick falls under the rules against irregular or dangerous actions, so expect at least a yellow and possibly a red, with the touch annulled either way. Covering valid target with another body part, or substituting off-target for target with an abnormal movement, is a yellow card and annuls your touch. Ducking and displacing the target are allowed, though. Deliberately hitting your opponent's arm off target is not an offense either. What is penalized is intentionally hitting the floor or anything other than your opponent to stop the action, which is a red card.
Body contact, called corps-à-corps, is not a fault in itself; the referee simply halts. A flèche or strong advance that ends in contact without violence is fine. Causing contact on purpose to avoid being hit, or jostling your opponent, is a yellow card with your touch annulled, and a collision with real force is treated as brutality and carded red. Turning your back on your opponent during the bout is a yellow card and annuls your touch. If you run completely past your opponent, the referee halts, and a touch you make after passing does not count.
Stepping off the side of the strip with a foot completely off stops the bout and costs you a meter of ground. If both feet go off, anything you do after crossing is annulled. Going off the side on purpose to avoid being hit is a yellow card. Crossing the rear limit with both feet is a touch against you, and being pushed off accidentally carries no penalty. If you think your opponent broke a rule, ask the referee one calm question after the call. Arguing a judgment of fact can earn you a yellow card of your own.