Its converting an encoded steam to HDMI output. There aren't enough pins on a lightning connector to directly output HDMI and even if it could, you still need a transceiver somewhere (not particularly trivial in terms of space or power to stuff in the phone).
The SoC is most likely a little ARM core to manage stuff (Cortex-M0/3) with a video decoder and HDMI transceiver, the 256MB of ram is there primarily for the decoder to use. It also needs to mux the audio stream into the HDMI encoding.
As for 'it runs iOS', your really splitting hairs, its pretty normal for the master devise to load a slave device with its firmware when plugged in (rather than storing the firmware in flash on the slave). The fact that they would use some knocked down version of iOS isn't terribly surprising. Embedded versions of more powerful OSs are used all the time; There is a non-zero chance your stove and microwave are 'running linux'.
MicroUSB+MHL accomplishes the same thing quite well on the current crop of Android phones. Why Apple chose the method they did is still quite odd.
Apple already made a strong commitment to AirPlay, so they already had a focus on building a fast, smooth, low power encoder that could encode the entire screen. Once encoded, the stream is probably only a few Mbit/s, a data rate that can easily be transmitted with single ended protocols like SPI. Almost all SoCs already have multiple SPI buses so no need to change any hardware.
Their solution may seem inelegant to some, but I think it is great. They managed to support a feature with almost zero hardware costs on the core device (depending on how you assign the lightning connector). A feature that I'm willing to bet only a very small % of all users will ever use. That is a big win, not having to dump extra hardware into a device that only 5% of all users will ever activate does great things for your margin and design flexibility.
Unless Linux comes with a BSD license, there is, in fact, zero chance. Apple is known to run a NetBSD variant on the airport routers - I'd say that is what is likely or whatever the hell a "stub version of iOS" means.
If you go here: http://opensource.samsung.com you can see that they have the OS for some TV's. This is the typical way they comply (with unuseful, unannotated giant dumps).
LG actually has some appliances:
For your two points:
1. I think that comes with the territory of being a Lightnin accessory and makes sense if they're doing some sort of compression or something before hand. It certainly fits with Apple's persona. Control the content, don't let any-old-body create a video adapter. They have to deal with Apple's handshake/stream/whatever because Apple doesn't pin-out the HDMI for the Lightning connector...
2. That is a major bummer and seems like one of those "details" that Apple would get right. I'm certainly curious the more I think about it as well.
I've tweaked the post to make it clearer that AirPlay isn't _necessarily_ the _exact_ mechanism being used! It could just be H.264 or MPEG or whatever.
Mostly I'd love to know exactly what this chip/system does, so if anyone here with far more advanced hardware knowledge than any of us have feels up to hacking around, that'd be amazing! :)
The key takeaways from the post are:
1. It's feels unusual that a AV adapter would have a full SoC ARM-based CPU with RAM etc, not just a video encoder/decoder chip. Is it unusual? Let me know. :)
2. It's a bummer that the Video Out isn't very good, and not true 1080p, and has MPEG artifacts.
(Thanks for sharing the link.)