The long white 'cable' is probably a tube with either a pull string for future fiber or already has the fiber in it. What does the writing say?
A splitter will attenuate signal level to/from your modem. Typically 3-7 dB, depending on the splitter. If you do not have a need to split, the coax, use a female-to-female feed-thru connector. If the signals are already marginal, don't use a splitter. Check the upstream signal level, before moving the modem. If it's getting above 42-ish dB, then splitter is a bad idea. The downstream signal level should be above -5-ish dB if you will add a splitter.
I haven't noticed the S33 getting as hot as some older modems, but I haven't paid that much attention. If it gets hot, then keep it in the open. Personally, I like to see all of the lights/LEDs.