Re question 1: when you learn this stuff in school you usually simplify the system by modelling it as a simple harmonic oscillator so the amplitude of the system will be given by some equation like:
$$ A(t) = A_0 e^{i\omega_0 t} $$
where $\omega_0$ is the natural frequency of oscillation. Typically you study what happens if you apply a force that also varies sinusoidally with time so:
$$ F(t) = F_0 e^{i\omega t} $$
where the frequency of the applied force, $\omega$, is not necessarily the same as the natural frequency of the oscillator, $\omega_0$. This is what your teacher means by saying that the force has a frequency - they mean the frequency $\omega$. In your teacher's example of a swing the swing has some natural frequency. If you are applying a force periodically, i.e. pushing on the swing in a repetitive way, then the force you apply also varies with time (though it is more like a square wave than a sine wave). The amplitude of the swing is greatest when the frequency with which you push the swing matches the natural frequency of the swing.
Re question 2: when you start learning this stuff you typically start with an undamped simple harmonic oscillator, i.e. the oscillator doesn't lose any energy. If you solve the equations of motion you find that the amplitude goes to infinity when the frequency of the driving force $\omega$ is equal to the natural frequency $\omega_0$. This is because you're putting energy in but the oscillator doesn't lose any energy so the energy just keeps growing.
A real oscillator like a swing loses energy through friction, and we call it a damped harmonic oscillator. The rate at which the oscillator loses energy is related to its amplitude, so as you push your system (the swing in this case) the amplitude increases until the rate of energy loss matches the rate you're putting energy in. So the harder you push your system the more the swing will move. In principle there is no maximum amplitude, though in real life there obviously is since at some point the swing will go over the top and start revolving instead of swinging to and fro. A swing isn't a simple harmonic oscillator! It's only approximately simple harmonic for small swing amplitudes.
Re question 3: Most objects will have a range of resonant frequencies called normal modes. However there are usually many normal modes and the frequencies of these modes are related to the object's shape in a complicated way. The Wikipedia article gives some examples of normal modes, or do a YouTube search for "normal modes" to find loads of videos on the subject - some really impressive!
Your understanding of resonance seems about right on a qualitative level. If one were to ignore losses like friction, drag, or the like, "driving" a system at its resonance frequency would indeed result in feeding it more and more energy which is stored in the form of a large amplitude of the oscillation. For a completely lossless system, the amplitude would grow to infinity. In reality, every system is lossy to a certain degree but for "high-Q"$^1$ systems (meaning the losses are small), the amplitude can grow so big that they rip the system apart. This is referred to as a resonance catastrophy and is what happened to the Tacoma Narrows Bridge.
Generally, the losses will be proportionally to the amplitude of the oscillation and also to their frequency. I.e. at resonance where the amplitudes are big, so are the losses. Another way to think about it is in terms of energy transfer. At resonance, it is easy for the object to pick up energy from the driver. The energy stored in the object grows at first, but it will eventually be balanced by the increasing losses. Instead, if the system is driven off resonance, it has trouble picking up energy from the driver, so the stored energy decreases (and so do the losses) until the losses balance the little energy intake that is still provided by the driver.
In summary, at equilibrium (meaning in the steady-state), the object will always dissipate as much energy as it picks up from the driver, otherwise its energy content (and thus the amplitude) would change. At resonance, it is easy for the object to do that, so there also the energy dissipation is greatest.
$^1$ Quality factor or $\mathit{Q}-\text{factor}$
Best Answer
Since you want to answer a small child, I'd just start with a weight on a spring, preferably with a physical demo at hand. Show how it bounces at a specific rate regardless of how you initially stretch it out. Then maybe show a different resonant rate when you change the weight, or length of spring. That takes care of the "natural resonance frequency" part.
Then, maybe a bit more tricky: gently tap the weight (vertically) at the resonance rate and show the amplitude growing. Then tap at some other rate and observe the amplitude collapsing or going pseudorandom.