This is a good a place as any to ask

I've added in a video for my rotation so you can see exactly how the rotation plays out in practice. It turns out I'm not very good at talking in videos quickly enough to explain the rotation while doing it, so hopefully the explanation here will be enough.
I use a Razer Diamondback (ambidextrous) mouse because I play left handed and so far I've been unable to find a left-handed naga (they're out of production

), so I only have a few spare buttons on my mouse and use them primarily for weapon swap and push to talk, which leaves my main skills to the num pad on the keyboard.
I don't block cancel on my stam sorc- I've tried it in the past on my bow bar and tend to find that it's simpler and smoother for me to let the animations play out rather than fumble through block cancelling. It's worth pointing out that I'm not the most amazing player when it comes to animation cancelling and weaving, so it's entirely possible that thoroughly practising block cancelling will get better results for you
Personally, my focus for weaving is to always hit the light attack in between all skills on my duel wield bar because they add a lot of extra damage there and it's a waste of crushing weapon if a couple are missed in a row. I'm less bothered about missing occasional light attacks on my bow bar because I prioritise getting the skills off smoothly so I can quickly return to my front bar. Most of my stats are higher on the front bar so the less time on the back bar the better, but light attacks are still useful for the hawk eye passive if you can get them in.
For animation cancelling, I always cancel the last skill before I weapon swap (e.g. poison arrow on my back bar). This is partly because it means the first hit of poison arrow lands while I'm on my front bar, so it does slightly more damage, but also because it's a good way to cut down the time completing animations without having to add in extra button clicks like with block cancelling. I also aim to animation cancel all my light attacks to save time, and queue up a skill to use after a heavy attack (always hurricane in the case of this rotation).
The main thing for me to build up to pressing buttons quickly was to practice really really slowly- I built up my rotations one step at a time by forcing myself to hit a light attack between each skill and weapon swapping in the right places. The damage was terrrible because I was doing it so slowly, but the aim was to get the muscle memory in place. I also practised animation cancelling with weapon swap, and animation cancelling light attacks with skills one skill at a time to make sure I had the hang of each element of weaving and animation cancelling before I added them all together. Putting weapon swap on my mouse was one of the biggest improvements for me to help the rotation flow and I like to think that having a proper left handed MMO mouse would help me hit significantly higher dps numbers, but I have absolutely no evidence to back that up other than vain hope
It's not the most exciting answer I'm afraid, but practice really is the key to efficient weaving and a smooth rotation

If I practised more I might be able to give you a better answer about block cancelling...