I don't think it works on AOKP or CM9, without installing an extra app. I've been using it on SHOstock 2 just fine. I think the TW ROMs have the support built in. I know 2.3 MIUI does not support MHL.
It only needs the power adapter if your TV does not support MHL. If your TV is MHL compliant it will send power over the HDMI port and charge your phone. The standard is still new, almost no TVs support it yet. You can still mirror the screen over HDMI with a MHL adapter, but you have to power the adapter. MHL compliant TVs should be able to control the phone through the TV, but because those TVs are as common as unicorns I haven't tested it. With a non MHL compliant TV all it does is accept the input from your phone. You must still navigate on your phone. But, you can use a Bluetooth keyboard and/or mouse to control your phone wirelessly.
I got the cheapest MHL adapter on eBay that shipped free from inside the US. I think I paid ~$7. Its worked fine for me over the past 6 months.
Sent from my Galaxy S II (i777)
> HTC 8525
> iPhone 3G
> iPhone 3GS (it was a dark time in my life, I was confused)
> Palm Pre
> HTC Aria
> Samsung Captivate
> LG Thrill
> Samsung Focus
> Samsung Galaxy S II (I777)
> HTC One X
> Samsung Galaxy Note II (I317)
> Samsung Galaxy S 4 (I337)
+ HP TouchPad