Skip BACKGROUND if you don't wanna know the background of the mechanics behind last hitting
Every hero has a last-hit animation
The base attack time is 1.75 seconds with base attack speed of 20. With that, check the charts on the Dota 2 wiki for the attack swing.All heroes have two portions of the swing
I call it the pre-swing (before the damage registers) and the back-swing (after hit recovery). Ranged heroes have different speed for their projectiles. The important thing is to understand that each hero has a very very unique animation to his/her swing and projectile speed and it takes time to fully understand it. For example,
Lifestealer has a VERY good attack animation, and so does 
Sniper, 
Puck and 
Queen of Pain. The pre-swing is minimal, so when you right-click on an enemy the hit registers quickly. On the other hand heroes like 
Spirit Breaker and 
Zeus /
Razor have hard to understand attack animations. The pre-swing is just too long, and/or the projectile travel is slow. This makes it hard to completely anticipate the last hit and get it right.Animation cancelling
So, when you are using heroes with less than desirable hit animations, it's vital to use animation cancelling because it allows your hero to complete the pre-swing before the actual attack even begins.To do it, you can leave auto-attack on or off (best to switch it off and only hit things you are SURE you wanna hit), right-click on the target and then press S (for stopping under the default keyboard scheme). Keep spamming this whole thing and only stop pressing S when you are sure that you'll get the last hit.
That answers your first two questions and your fourth question.
You can cancel animations with every single hero, but sometimes you don't have to. You usually use it with heroes with horrible animations, or when you have insufficient damage to out-lasthit your lane oppoent.
As to whether you can compete without this technique... well, to answer your question honestly, it's a big NO. I used to think that I can manage without, but after a few tough lessons in mid, I no longer think it's possible. It's very convenient to abuse since you can bluff your opponents into thinking that you're going for the last hit, as well as ensuring that you virtually have no pre-swing. Sometimes I target the creeps too early, so anim-cancelling bails me out. Just as well that sometimes I'm out of range so I need to get back into range and once the attack starts but the creep isn't ready to die yet I have to anim-cancel.
Learn it and get used to it. It's pretty necessary.
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