You sound like a WP neophyte. The Action center was a much needed feature that WP needed, but the live tiles provides an experience that neither iOS or Andriod can. It's always been one of the biggest selling points for WP. I've always found customization on Apple and Android devices limted because it doesn't really encompass the owner of the phone. So you can change the background...big deal. You still have the same small icons. You can't change the sizes nor change how they display information, because there is only one way. I don't have to touch a thing on my start screen to see who sent me a text, or create a shortcut of friends, co-workers, or family members, then have those shortcuts consistently update me with the latest posts, preventing me from having to navigate through the native apps like facebook.
Having all your mail linked together was a feature of WP8, so obviously you're not familiar with how to set it up. I have my outlook, hotmail, and yahoo all linked together. It's fluid and smooth.
My phone updates apps automatically, so again, are you sure you have 8.1. I agree that the shutter speed on the 1520 is a little slow, but like you've mentioned, it takes some of the best pictures ever, which is a given being it's a Nokia. Cortana is purely Beta, so I expected a sluggish experience. It's Siri and Google now combined, and is extremely inquisitive. She can integrate with Xbox music, Nokia Here applications, activate you Quite hours, which is another awesome feature you never mentioned. Data Sense, Wifi Sense and battery life are awesome additives that iOS doesn't have, and Android isn't as detailed. Not only does it show you what your battery life is, but more importantly what is draining it. The People Hub is something that my iPhone and Android friends love.. One consolidated spot where I have my contact lists, can see live feeds from Facebook and Twitter at the same time. Integration on another level. Driving mode, a WP exclusive.
As for the Galaxy Note 3, the fact that it has the S pen is the only significant upper hand it has over the Lumia 1520. You might need to take yours back to the store. It has the biggest battery with the best lifespan. Cortana shouldn't be sucking it down like that.
Allot MS the time it took Apple and Andriod to cement themselves where they're at...the fact is that you won't need to because they've already covered more ground in WP's short lifespan. Not only is the WP ecosystem on par with it's far older, maturer counterparts, but in some ways it's better, and it's only going to get better.