The CPU is fine. But note that even though our library, BiblioPixel, makes it seem like you are just updating a group of LEDs that in fact every time you call update it pushes *all* the pixels again to the whole strip. That's just how they work. So there's no difference between updating 1 pixel or updating all of them as far as the AllPixel is concerned.
LED change speed is dependent on the types of LEDs you have. WS2812 are the slowest, LPD8806 or APA102 are fastest. 700 WS2812 pixels like you have will take ~24ms to each frame, meaning a max framerate of about 45fps. This is just math... WS2812 has a datarate of 800kbps and you've got 700 pixels each with 24 bits of data per color for 16,800 bits per frame. The math here actually says 21ms per frame, but there's a ~2-3ms overhead of sending that data over USB first.
As for the rest service, I honestly recommend writing an animation class (as I linked to earlier) that runs in a thread (just use threaded=True in run()) and then send data to that animation when you get the request over the REST service. If you give me a little more detail as to what exactly the effect you are going for is I might have something I can send you that would get you much of the way, but would have to wait until this evening anyways when I have access to that code.
24/7 is no problem. I have multiple permanently installed displays, one of them on my porch in a weather proof box but otherwise at the mercy of the cold. They run all day every day with no problem. Even when off. I actually have them setup to keep setting all pixels to off ever second when no other animation is playing (this is just part of PixelWeb). This may sound a little weird, but when you have really long runs of these LED strips they are susceptible to RF interference, even to things like turning a nearby ceiling fan on, and then you get a few stray pixels turning on. Having the pixels continuously updated takes care of this. But point being that it's being doing this for years now.
I almost exclusively use MeanWell brand power supplies like this one: https://www.amazon.com/Taiwan-meanwell-switching-LRS-350-12-monitoring/dp/B00YMA0M6G/
You can find them on Amazon, Mouser, ebay, etc.
You have to wire in your own power cord and note that most come switched to 220V so be sure to flip that over to 110V if you are in the US. But they are the best watts per dollar you can get. Given that I have displays that run 500W @ 5v (100A!) these things are great. You can also get them in 12v and 24v variants if your LEDs require it.
Not really sure what other materials you need... but pretty much all wires, connectors, etc I either buy on mouser, ebay, or Amazon.