You've got it backwards. Interlace runs at 60fps, progressive runs at 30 (or 24) fps. You're right it's not technically 540p, but in 1080i there are only 540 lines displayed at any given time (hence the double frame rate, which is required to deliver the two half-images at 30fps),. Also keep in mind that your HDTV is not natively capable of displaying interlaced video (interlace only works natively with CRT displays). If fed interlaced video it has to run it through a deinterlacer, which degrades the video, since one interlaced frame is actually comprised of two half frames shot at different moments in time. If you were to simply combine two consecutive interlaced half frames into a 1080 line still image and look at it there would be motion blur (combing) on anything that is moving. So the deinterlacer has to 'interpolate' (fake by guessing) the other half of the information that it thinks should be part of each progressive frame.
1080i upconverted to 1080p will never look as good as something shot and displayed in 720p. It's actually a bit smoother to downconvert to 720p from 1080i because there's a lot less missing information that the deinterlacer has to interpolate to make each progressive frame.
No, I don't have it backwards. The 720p official HD specification is 1280 x 720 at 60 frames per second. Yes, there are 30fps and 24fps specs, but those are not what I was referring to.
As to how many lines are displayed at a given time, that is dependent upon your TV set. On a CRT with a longer decay, a 1080i image will have all 1080 lines displayed all of the time. It is true that alternating lines will be refreshed, but the "previous" line will still be displayed all the way up until it is refreshed again. They are not flicking on and off instantaneously.
As to which will look the best at various resolutions, that is almost impossible to say because of the compression being applied. The compression is not a constant and thus the quality of the image will always be varying.
Oh, my HDTV can natively display interlaced video because it is CRT. What it cannot do is display progressive video.
A full bandwidth 1080i signal will contain more digital information than will a full bandwidth 720p signal. However in practice, who knows how many bits are being delivered on D* and E* channels from hour to hour. A 1080i channel receiving more bandwidth and therefore passing more detail could easily look better when upconverted to 1080p than a 720p channel which is receiving less bandwidth.
Heck, you aren't even sure when the providers are passing along a full resolution image. E* has downrezzed to 1280x1080 on some of their 1080i channels in the past. Don't know what they are doing now.