TLDR: Why not get a true sinewave inverter?
Possible but truly wasteful. You would save a bit of money just going for a true sine wave inverter in my opinion...
Definite nope, a capacitor is like a blur tool. It makes edges rounder but it doesn't change what is essentially a square. The edges of the square get slightly more round but it still has a flat top and flat sides (mostly). Even really, really big capacitors (in terms of farads) won't make this look like a sine wave
Definite no.
"Modified sine" is a fancy word for pulse width modulation. The output a square wave and the size of the "square" varies with load and input voltage.
The total harmonic distortion, for all intents and purposes of our discussion, is simply a measure of how close you are to true sine, so naturally a modified sine will score very badly on such a test.
Square wave will still be square wave after a transformer. You can think of a transformer as follows:
Two buckets of water. We call one bucket the primary and another bucket the secondary.
The primary bucket and secondary bucket are always the same (due to magic, called magnetism). So if you fill up the primary bucket, the secondary bucket will have the same level of water! Magic! Some buckets take more water, some less. So think of the water as voltage. But their LEVEL is always equal even if the amount of water they have isn't (so voltage can differ but it is at a ratio). The way we fill up these buckets can change: Therefore sine wave is a different way of filling the bucket from modified sine.
This is how sine wave works:
We fill the bucket slowly and the water level rises in both buckets gradually and then drains again down gradually. This looks like a beautiful wave, like you would see in the ocean. It raises and falls gently.
This is how modified sine works:
We suddenly have a full bucket that stays full for a while, then suddenly it is empty again. There is no in between gentle rise and fall. Because both primary and secondary are "mirrors" of each other, the level of the water is the same, it is just the amount of water that can be different (what we think of as voltage).
Because both buckets are the same level at all times, the wave doesn't change after we suddenly put it through the buckets.
Does this metaphor make any sense? It is simplest way I can think to explain the function of a transformer. It is obviously not that simple because the change in a magnetic field is actually what induces a current, but this explains the idea behind a transformer more clearly I think. (Therefore modified sine saturates the core instantly, then burns some current as heat, and the magnetic field collapses suddenly whereas sine wave has a gradual rise and fall. In both cases the magnetic field induced in the secondary will look the same. There is no way it will suddenly change its shape except that it may have a shorter width due to the core saturation in the modified sine case. Which is why your electronics run at much lower power and the core saturation means lots more heat)
Edit: If you are really interested in how this stuff works, the electronics we are most familiar with we know as the electronic circuit, which we use Ohm's law and etc for. Every electronic circuit has a
magnetic circuit. Transformers operate on this magnetic circuit. Every formula we have like V=IR, P=IV, etc. has an equivalent magnetic circuit formula. There are also rules in the magnetic circuit, just like the electronic circuit. Very fascinating read and must learn for anyone that truly wants to grasp electronics.
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