Yes, it'll definitely connect, and there are a few options that I know of:
Write an iPhone app - but this could be a steep learning curve and you need a developer subscription
Make it into a Bluetooth keyboard - that'll work fine to control an iPhone, but I'm not sure you can get any useful data back out of it
There may be an existing app you could use...
I don't know if something like Blynk can be made to do what you want. If you're interested in this let me know as I've intended to figure it out and do a tutorial for a while.
The Adafruit Bluefruit app can talk to Espruino pretty easily, but I don't know if that'll integrate usefully with iOS notifications.
Use the WebBLE app which implements Web Bluetooth - so you can write a webpage that'll then control the Pixl. I'd definitely recommend that as it's really flexible, but it's not something that can run in the background so wouldn't do notifications.
Espruino is a JavaScript interpreter for low-power Microcontrollers. This site is both a support community for Espruino and a place to share what you are working on.
Yes, it'll definitely connect, and there are a few options that I know of:
Hope that helps!