Okay so this might get a bit repetitive but what I'm saying is important. For starters, you should always ward the bushes or river while waiting for gromp/kruggs to spawn. These wards will keep you safe in the case that the enemy decides to invade. You got lucky this game but I noticed that neither side warded river. If the enemy invades and you don't ward you'll be put incredibly far behind and giving away the information that none of your team likes to ward. Next, you decided to clear buffs last? I'm not entirely sure why you did this. Even if you're playing a manaless champion like Zac you need to make sure that you clear buffs early for the experience, gold, and resources (blue buff doesnt just give mana it also helps with cooldowns).
After your first gank where you killed Kog'maw, you should NOT have done dragon. I'm sure you figured that out but here's why. One, you had no wards behind the dragon pit or anywhere in the river so you had no idea if any enemies were near. Two, both you and lucian started out incredibly low. Lucian probably figured you weren't going to do dragon because of this and came in late, forcing you to take more damage. This simply means that you need to be more clear with your intentions so that Lucian comes sooner (drag was bad either way). Third, the way you did dragon was weird. You and Lucian stood clumped together so that whenever dragon attacked, both of you were taking splash damage when only one of you needed to tank damage. Overall, your idea was in the right direction. "Successfully ganked bot and killed Kogmaw so we should secure dragon". The execution was just a bit off. Rewatching I feel the biggest problem is still a lack of wards and not having everyone on the same page. In lower elos nobody will be on the same page so you either have to ping constantly or make sure things are WAY safer before doing objectives so you won't be punished as badly for your bot lane being so slow to follow up. (On a side note, by the time you started drag, bard and kog were both up and on their way to drag. So a bit slow in starting drag. Lucian and Sona both should have focused drag when it was that low so bard couldn't steal it as well.)
I just noticed that someone said you screwed up by letting Kogmaw and Malphite through. That's not necessarily a mistake as there are so many champions that need bans right now but w/e. You picked Zac which can actually counter Kog'maw simply because he's a champion that can get to the backline with a good slingshot. I haven't finished watching so I don't know if you capitalize on this. Just know that Kog'maw is very vulnerable to champions that can get to him and you are actually the only champion on your team that can do this consistently. Annie would require flash, sona would require flash, lucian has a short range, lee would require flash or mechanics you just won't find in bronze. The rest relies on Kogmaw being out of position.
When you went to gank top, you stopped to kill the ward in tri. You're ganking and have already been seen so you're wasting time at this point. The ward kill just lets Volibear catch up and Lee to take more damage. If you wanted to gank, then do it and don't hesitate. Also, ping before you engage. Multiple times if you have to to make sure that your team knows what's going on. This is more of an overall communication note
Now, I don't play a ton of Zac so I don't feel confident talking about Zac specifically. There are a few notes I will make just to get them out of the way. I often see you set up your slingshot when you're nearly melee range. After that gank top, you moved midlane. You were backing off when Annie engaged and you realized that you could actually fight. It's at this point that you start to use your sling shot when your in the bush and nearly in melee range. Here you should have just started smacking Kassadin and just saved your sling shot in case you needed to chase him down because now you have no way to do that. When you use your sling shot to chase, you can fully wind it up and go an incredibly long distance.
Right after this (12:11) you decided to tower dive because you saw Kog'maw return? You did this when the minions were in the middle of the lane and with no pings or anything to alert your team. Because of this, they followed up late and this led to lucian dying under turret. The most important thing to note here is that you should have pushed the wave into tower before diving. You take less aggro from the tower and minions this way. You survived because you're a tank and you managed to get out of tower aggro range. However, once you left, the aggro switched to lucian which is how he died. If you'd have waited, the dive would have gone faster and smoother without Lucian dying.
I have an exam in like 30 minutes so I'm going to stop here. I might finish watching later, but I'm not sure.