About the original question regarding implications of having the friend register someone else in his HHR:
From reading some of the application forms, it seems that there might be a minor implication that if the only people on the HHR are you and your friend, and your friend later moves to a new address (either within Taiwan, or overseas), then you might become the “head of household” of the old address, unless you also move your HHR to a new address.
On a slightly different topic, following up with tentative answers to my other questions posed above:
The page I linked above (https://service.gov.taipei/Case/ApplyWay/202112270006), describing HHR move-out registration, says: “持我國護照或外國護照出境2年以上應辦理遷出登記；出境未滿2年，亦得依當事人出境之事實辦理遷出登記。”, which Google translates as: “If you have been out of the country with a Chinese passport or a foreign passport for more than 2 years, you must apply for move-out registration; if you have been out of the country for less than 2 years, you may also apply for move-out registration based on the fact that the person has left the country.”
This implies that you might need to actually leave Taiwan for some time – in order to establish the 出境 status of leaving Taiwan – and only then are you allowed to apply for move-out registration for your HHR. It seems it should be possible to do the move-out registration online, without needing to return to Taiwan, at this site: 中華民國 內政部戶政司 全球資訊網 - 線上申辦出境遷出登記). (Maybe a TECO overseas office could also help, though I couldn’t find specific evidence to support this idea.)
As for the other issue about whether or not the HHR office keeps track of your foreign address after moving out:
It seems that the application forms for the HHR move-out registration – both online and paper forms – only ask for your phone number, and possibly email address (when applying online). There seems to be no way for you to report what your overseas address is, implying the HHR office, and other Taiwan government offices, don’t know and don’t care about your overseas address, so when living overseas, you would need to be proactive and reach out to the various Taiwanese government offices to check if there’s anything you need to do for or pay to the Taiwanese government.
At some point I’d like to make a wiki post here on the forum about all of the various obligations that are incurred when you first establish HHR. Items I’m aware of are national pension payments, NHI payments after qualifying for NHI after 6 months, and 31-day tax residency rules when you have HHR – but I guess there may be other obligations to be aware of also.