Are we talking horizontal lines that are slightly diagonal that run down the entire screen? If so, these are called retrace lines and are supposed to be blanked by the video IC. This is usually caused by either one of three things, first, it could be a heater-to-cathode short, but since they are white I would say it is probably not because shorts are rare and occur usually only on one or maybe two heaters (electron emitters), it would take all three to create white retrace lines). Second would be that the grid voltage is two high on the CRT (otherwise known as the G2 voltage). The third possibility would be that your RGB cutoff voltages are way too high (somehow they were reset), or a resistor or some current limiting component failed on the CRT neck board creating a larger than normal voltage to present across the filaments.
I think the most likely cause is a failure of either the EEPROM that controls your RGB cutoff voltage or G2 or "Screen" voltage to rise way too high, or a failure of a component on the CRT neck board that either allows the G2 voltage to rise considerably or the component that blanks the screen during retrace on the video IC has failed. My guess would be that has more to do with whatever EEPROM or static-dynamic memory that is inside the monitor and TV (which controls many of these voltages, on newer TV's and monitors) has either been partially corrupted, or reset to some sort of standard value that is causing you to see the retrace lines. Then again this would all be a moot point of they are not retrace lines at all. Anyway, good luck with fixing this, seriously, it can be quite tricky and expensive to diagnose and fix CRT's problems these days.
I only skimmed through your post right now (I don't really have time to try to figure out everything you said right now, but I will try tomorrow... I don't really know much TV... uh... jargon though >_>), and I could tell you right now that the lines don't seem slightly diagonal, atleast not to the naked eye. They look completely perpendicular to the vertical lines of resolution. I should also add that what I really want to know most is if it would be safe to put my new TV on that circuit. Barring some sort of divine intervention, we won't be getting this older TV fixed... now that we've bought a replacement for it, it's not worth paying probably a few hundred dollars for a repair. Do you think that something like this is likely to happen again, or is this sort of thing usually an isolated incident? Would a basic surge protector ensure the TV's safety? The damaged TV has always been lead through a surge protector, but I'm not sure if it was even functional when the TV was actually damaged (I thought I read something about surge protectors becoming defunct after roughly 2 years... I could be making this up >_>).