For starters, keep track of enemy cooldowns. For example, if an enemy Lucian has just used his Q to farm minions, you know that his primary damage spell is down for like the next 6 seconds and that is your window to engage. However, mind the minions; especially during early levels the casters will do a lot to you and might make you lose trades, so make sure to kill the minions first.
Furthermore, take note of your own minions' health. This way, when you see a minion on low health, chances are the enemy AD will try to last hit it. During this time, harass them with an auto or two because they can't fight back while csing.
As for Azir, positioning is key. Stay back during teamfights, and always watch where his soldiers are at. I'd say if the enemy team picks an Azir, you should pick an AD with a dash of sorts; Lucian and Graves are great starters (I'd say Graves more as Lucian has fallen off a bit).
Sivir is a teamfight AD. Her damage isn't very high in small skirmishes. She thrives in teamfights as her W does a lot of damage. I wouldn't worry much about the amount of kills you rack up in a game, as long as you are dishing out damage and get kills for the team. Since Sivir has a low range of 500, always watch where you are standing relative to your team and the enemy team. All you've got is your Spell Shield and
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to save yourself from that one positional error.
Finally, keep practicing! I was Bronze II in Season 4, but I kept practicing and playing games as AD. It took me a while to learn how to CS properly, how to kill the cannon minion without using an ability, and most importantly, position well in fights, while dishing out tons of damage. And here I am now, sitting in Gold 4, after a years' hard work.
Last note: BUY WARDS. VISION IS KEY.
Best of luck on the Fields of Justice!