Can I rip a DVD that is 60 fields per second to 60 frames per second?
Note that a 1080i60 video would have 60 fields per second, not frames - each field consists of approximately half a frame (or rather, every other horizontal line of pixels interleaved). Unless you are viewing the video natively on a CRT-based display, it will probably have to be deinterlaced before being displayed regardless (in which case, it will be down-converted on-the-fly to 1080p30).
Youtube videos are not always 30 FPS, it can differ. It is indeed, however, progressive, as there aren't really any interlaced displays anymore. Also, x264 appears to be able to output interlaced mode should you desire it.
However, when you watch content on your TV, I'm guessing the TV converts it to progressive for you, but the end result is a very fluid picture that "feels" quite a bit like 60frames/s.
Regardless, 1080i60 will still appear to be "jerky" under fast motion. 1080p broadcasting standards have been ammended to include 1080p60, which would be especially visible under sports and fast-motion videos. Many DSLR and mirrorless cameras can also output high-framerate progressive video now.
Furthermore, the "fluid picture" might actually be a side-effect of any motion interpolation (e.g. Motion Flow, TruMotion 120Hz) which often leads to the "soap opera effect" and other visual artifacts, although this is also highly content dependent.
TL,DR: I would convert and encode the video as progressive unless you had a reason to keep it interlaced (e.g. editing source video), in which case I would probably opt to just keep the source medium as-is and not recompress it. Keep in mind that almost all new displays are purely progressive in operation, as only scan-line displays are capable of displaying interlaced content.
You can apply filters to reduce deinterlacing noise as well, which would keep the viewing experience consistent between computers (which might have a different filter/output stack and deinterlace differently). Using Handbrake, you can deinterlace or decomb the video before compressing it.