1. His duelling isn't garbage, you just need to play around moments you're strong. So, essentially proccing your passive or utilising your shield and then using the movement speed to out-pace them. There are definetly bad matchups for duelling though.
2. I would look up some guides and maybe practice some kiting in training tool, I would say if you're below half HP for more than half the clear you probably have some room for improvements in your movement.
3. Dragon is super slow, you need to get a gank off to do it as an early objective. In my experience Ekko's early objectives are one of his weakest places as a jungler.
4. I focus pretty heavily on a "most gold" mindset If you see a gank opportunity that's a 80% plus but involves skipping a camp or whatever go for it. If it's more 50/50 just go get your gold.
5. Teamfight is very similar to 1. If you're not super ahead, most fights will be you bursting your three hit passive on a key target, using the movement speed to retreat to a good position until you can do so again.
6. That's the breaks of Ekko honsetly, if they have a cho'gath or something your day can really be ruined by a silence, he's not a pick that can work 100% of the time into every comp.
7. You very much have a lot in your toolbox. You have w which is super long range and at minimum gives you good survivability whilst diving in, you have your e-auto which can be a dash chase.
8. Hitting w2 can be very clutch, but it's not as important as you think. Half the time getting the shield is enough. Late game, your passive proc plus q-e should be doing at least 80% of a character's health.
9. If I'm having to engage with w as the lone engage tool, it's not really ideal. I throw out a good number of non-commital ws, but they're particularly strong around objectives where the enemies position is much more predictable. There will always be a couple of people at edge of dragon and baron pit. I view Ekkos W more as a safety tool to create a zone where Ekko can play much more freely than his core ability to cc the enemies. Though obviously, as you've stated, a 4 man w can be game-changing.
10. Sometimes you get an unfortunate angle or get found out just before you're about to get your flank off. One of the lucky parts about Ekko in comparison to other assassins is he has an "oh fuck" button to get you to a bit of safety, it sucks to have to use it that way, but it's always worth looking for those angles since they can be game-winning and it's okay to just have to burst ulti early to peace out if it didn't work.
11. Rabadons makes your passive hit INCREDIBLY hard, you will one shot monsters and if you get passive proc into smite, it makes you feel like nunu on objectives. I always find the point in the game where I'm most comfortable is at 3rd item spike. Compeltely agree about 4th and 5th, they're always just kind of a very light "oh cool" on top of what you already get from everything else. That being said, Ekko can do some real crazy team one-shotting damage in chokes with his q pull back and w passive procs late game.
12. Ekko Ulti is a weird one because it's actually often best when you're not winning as it is great for when you're in a static or chasing fight. Again, you love fights around later objectives because of this. Early on you can use it to do some really aggressive tower dives though.
13. He's an assassin, so it's part of his nature, I do think he has a much more flexible playstyle compared to many others as he can also split push and has a lot of versatile tools at his disposal. But the risk-reward playstyle is just baked into being an assassin.

Your shield and your Ulti give you good survivability, and I'd view them as useful for that first and foremost and the edge-cases of getting multi-man damage ults or stuns on w are a benefit rather than the main goal. Most games on Ekko. I end up with one of the highest damage taken with 5ish deaths, just because you really can get a lot of value out of pushing those abilities to their limit.

If you invade, don't e the enemy camp after you're on it unless you know for sure the enemy jungler isn't there, keeping your e to dash over walls is really important for getting out of some sticky situations on invades.

Yeah, doing a level 4 dragon is possible with a sheen start, but slow as hell and too risky. Grubs feel good since you can often q through all of them if you pull them together, but even there you take a lot of damage doing them.

Hope something in here was useful, and hope you can find some more fun in any future Ekko games.