First of all, a wave is something that propagates and bears some periodicity during its propagation. Mathematically, this can be elegantly described by a certain set of mathematical formulae, completely independent of the physical meaning. So, there are many similarities between, e.g. sound and light waves, but also differences that are unique to each of those physical phenomena.
What is the difference between sound and light waves and why does one require a medium
Light/electromagnetic radiation is a form of energy of its own. Once "created", it can exist on its own until it is converted into another form of energy, e.g. when the wave is absorbed in matter. A sound wave on the other hand is a periodic motion of matter (air when you speak, water molecules when you think of the noise from ships' propellers, iron if you think of the vibrations you create by hitting an anvil with an hammer). If there were no molecules, an initial vibration would not move anything... and therefore there wouldn't be a sound wave.
Sound is a periodic density modulation of matter. Therefore it needs the matter to exist and the matter is the medium that the wave traverses. Light is a form of energy that can exist on its own. Therefore it does not need a medium to exist.
Why can electromagnetic waves travel in a medium
The answer is complicated, since electromagnetic waves cannot "just travel in any medium". You can test this by yourself: On a bright, sunny day, place your hand before your eyes and turn your face right towards the sun. Can you see through your hand? No, you can't. Therefore, the light from the sun does not travel through your hand.
Light as an electromagnetic wave can be affected by electromagnetic "objects" - and those exist plenty in any form of matter, because molecules and atoms are made up from electric charges. Those electric charges can be combined in many different ways. Some of them allow the light to pass through, some of them don't. To put things in very simple (but hopefully better understandable) terms: Typically light bounces off of molecules and around in the matter. Due to this bouncing, the light is slowed down, i.e. the velocity of light is lower in matter than in vacuum. Another thing that happens often is that light just has the right frequency and all its energy is gone after it hits an appropriate molecule. The energy is then converted into vibration and rotation of the molecule - and the light itself is gone and can therefore not pass the matter any longer. This is what happens in the little experiment proposed at the beginning of this section. If you happen to have a matter where light can just go through, then you could think of the molecules as receiving antennas which are not tuned to the correct frequency. The light wave does not "see" those antennas and just passes between them (and there is lots of space between molecules).
So, light can travel in matter only, if the matter physically allows for that. Typically, it doesn't. Light can be absorbed, reflected, or transmitted. It depends on the kind of light (i.e. frequency of the electromagnetic radiation) and the kind of matter that it interacts with.
Are all electromagnetic waves quantized?
Yes. They are. Also heat radiation. Heat is a critical topic, since the kind of heat that we humans can feel is "incoherent motion of molecules", which can be mediated by matter and radiation. The latter one is the one you were asking for.
Yes, they are.
Can waves of each type have the same wavelengths?
Yes, this is entirely possible, but some physical limits must be respected. I.e. it does not make sense to have a sound wave that has a shorter wavelengths than the size of the molecules that it its medium is made of. Sure, there are other limiting factors in the sound wave example that have to do with inertia of molecules, viscosity, etc... But let's say you have a sound wave with 440 Hz... It is no problem to have an electromagnetic wave with this frequency. You just won't see or feel it, because of its low frequency, that is well below anything we humans can perceive.
Yes, it is possible, but don't mix up the effects - you would not start to hear such a light wave, nor would you start to see the sound wave.