Yep, SMS push. I did some prototyping using this tech for a telco company here in Sydney back in, like, 2005 (9 years ago!). It actually worked quite well.
No, but it is not uncommon for J2ME apps to have a different package for certain vendors, or even for specific series or models of certain vendors. Those packages often utilize vendor-specific APIs or work around limitations or quirks in their J2ME stack. Opera's J2ME application took this to extremes, with dozens of different versions of a specific release.
I'm surprised - I wrote a Twitter client in 2009, and I couldn't find a way to make that work. I didn't think J2ME could do that at all, to be honest.