@P5music: That first link recommends installing Ubuntu or Xubuntu as a base system to cross-compile stuff for the eeeNote. Right there is a significant point of impedance if you are asking the TinyCore community for help.
It's great that you think highly of TinyCore and want to see it on other platforms such as the eeeNote. I do too, but IMO its not ready for such a move right now. While it does not hurt to ask if there is someone hanging out here who has the time, skills, and interest to help you with a port to eeeNote, don't be surprised if no one offers to do it for you.
"Guys on that thread seem not to be afraid"... I wouldn't say it's fear that makes interest here tepid. More of a matter of value obtained per time invested. I'm not sure you really appreciate how big an effort it is likely to be. My 20+ years of programming (including many years cross-compiling) leads me to estimate it could easily take a few hundred hours to port TinyCore to ARM and this eeeNote platform, assuming the person is already reasonably competent and knowledgable. Still interested? I'm guessing that the number of people other than yourself in the the TinyCore Forums who have this much available time AND have an eeeNote to work on is awfully close to zero... but who knows, you might get lucky. Perhaps it is easier to get people on the other forums catering to the eeeNote to use some ideas from TinyCore to make their Linux stuff smaller and faster.
My advice for a more realistic approach: take what those other guys are doing and install it on your eeeNote. Study it closely and try to find out how to make it more efficient by removing things you don't need or substituting in smaller & faster alternatives. Maybe you can try to substitute a few things found in TinyCore after you learn to recompile them for ARM. Or get those other guys to recompile a few apps from TinyCore for you that allow you to slim down that system. With this approach, you will be learning how to tinker with your eeeNote. If you get to the point of easily porting various apps to it, you can then try to port all of TinyCore. I'm sure you will get help if you ask, as long as you are doing a lot of the work yourself.