This was very good stuff. I enjoyed watching these.
As for Lucario's forward air to down air bit, the most you can do is get good/lucky DI on the forward air to escape. You can also double jump through the down air sometimes, but I don't think it's worth it. Running away is the best idea, but even that's risky since Lucario can just follow up with Aura Sphere spam. It's tough, but I'm sure if you keep an eye on when that particular approach is used with Azen's playstyle, you'll find a way to escape it that makes you safe.
As for down air in general, if it's being used from directly above (most likely after a tilt or a throw), you can space it really well with up smash.
Be careful. Remember that Yoshi jumps during his upsmash, so try to release it just as Lucario's second foot is going down so you can catch him in that sweetspot between consecutive down aerials. You can bait down aerials with up throw much more than down throw, but at lower percentages, this is obviously a bad idea.
Besides Azen's spacing and his myriad down aerials, he rarely seems to use it when it seems like it could be baited, which is tough. I find the best or easiest way to fight Lucario is to know what move he's going to do when you want him to and react accordingly since it's very hard to meet him head on. With that said, try to spot out when he feels comfortable doing certain moves (as noted by posts above) and capitalize on that as much as you can.
I hope these suggestions don't seem like cop outs since the advice isn't very specific.
I really enjoyed these videos. Keep it up.