As stated above. Does expending a ki point to add extra damage to an attack you make count as "spending ki as part of your action" allowing you to make an additional monk weapon attack as a bonus action. So, basically at level 3, you could either make a monk weapon attack and then an unarmed attack as a bonus action OR make an unarmed attack, spend a ki to give more damage, and then make a monk weapon attack as a bonus action. At the end of the day, you still get to make the same combo of attacks, with the ki only determining if you apply extra damage or not.

If you CANT apply ki fueled attack this way, then the only major difference is it would become unarmed attack plus damage followed by another unarmed attack as part of your Martial Arts

Looks right to me. Ki Fueled Attack means that spending ki as your action for any reason allows the BA monk weapon attack.

Going unarmed strike first does add risk though -- you need to hit to be able to use Hand of Harm, so in an instance where you do not hit you cannot spend ki to proc Ki Fueled Attack. Whereas monk weapon then martial arts BA will always work (of course, you can always unarmed strike then BA martial arts as well, whether you hit or not).

Yes this should work. At 5th level, if you unarmed strike first and hit you can apply the HoHm, then attack with a weapon as your extra attack. Then BA weapon attack. (1 unarmed strike, 2 weapon attacks)

If you miss the first unarmed attack you still have a second chance to hit then BA weapon attack.

If you miss with both unarmed strikes on your action you still get your BA unarmed strike.

Ki fueled strike does come in handy.