How to check (with DIY methods) if an Inverter returns a Square or a Sine Wave?

I would try a soundcard. Connect a resistive load (eg. light bulb) to the inverter. Wrap a piece of wire around one of the cables leading to the bulb (DO NOT CONNECT DIRECTLY), connect to mic input and try recording. You should be able to see how (un)clean the power is.

One way is to find the difference between peak and average of the half-waves.

Use a diode full wave bridge to rectify the output of the inverter. Put a 1 MΩ resistor across the output of the full wave bridge. Measure that with a ordinary voltmeter.

Now add a 10 nF capacitor across the resistor. This cap should be rated for 1 kV or more. Such caps are readily available up to 10 nF or so. Measure again with the meter. If the voltage is basically the same as before, then the output is a square wave. For a sine wave it should go up substantially, like 30 to 40 percent.

With a high fps camera set to manual exposure, a big resistor (>~10k), and a couple of scrap LEDs (not white) back-to-back in parallel you should be able to measure the brightness -- does it change smoothly or in steps? But you are working with mains if you build something like that, even if it is isolated from ground.

How easy it is to read the brightness of a spot from a video is up to you (I've written Python to do this for a series of stills but never coded any video analysis).