Waveform not reflecting multitrack edits

Yes this is all normal. You aren't actually 'editing' your track as such - you are merely selecting which bits to play at any given time - that's how non-linear editing works. To store that as a permanent result, you either have to export, or mix down, your track. When you do this, your edits will be reflected in the result. The differences between Multitrack and Waveform playback can be caused by several things - anything you do in the way of effects in Multitrack won't be reflected in Waveform playback of the same material, but the one that most people notice is caused by the way that stereo panning is selected.

 Non-linear editing has several advantages. You don't alter the original material, so you can always go back and alter things, and it also makes 'versioning' a lot easier. And obviously you can layer tracks - which you can't do with 'conventional' editing.

The best way to think about all editing is that it's a process of selecting what it is that you want to include in the final result - it's not about cutting things out. And that's what happens in Multitrack - you are arranging the bits you've selected and playing them at the time you want. If you cut stuff up in Waveform view (which of course you can, if you want) you have to remember that if you save the changes, that becomes permanent. Which is why you should always work with a copy, not the original file.