Hi, ledojaeger:
If you're a native speaker of English, you can use "yaru ki" similarly to how you would use "feel like...," "bring oneself to do...," "be motivated to do...," "be interested in doing..."
To answer your question 1):
Most frequently, I see "yaru ki" coupled with "nai" or "denai." (see 3 below for some positive constructions)
Yaru ki ga denai (I don't feel like/not motivated to do [something])
Yaru ki ga nai (They/he/she don't seem to be interested in (willing/motivated) doing [something])
Here's a good example. My company took over a big chunk of work on a project from Company B at the client's insistence. Company B was still doing a lot of the project, which they had to coordinate with us. They were sloppy, slow to respond, complained, and did the bare minimum to not get completely fired. So...we used this construction quite a bit around the office after our interactions with them.
Co-worker (angry after talking to Company B on the phone): "Mou, mattaku yaru ki ga nai ne, ano kaisha."
To answer question 2)...I'm not a native speaker, but I don't think you'd use a construction with "yaru ki" to say you need to motivate yourself. Probably something more along the lines of "ki wo ireru," "ki wo mochinaosu," "ki wo hikishimeru," or maybe "kiai wo ireru." I'll defer to a native speaker here.
To answer question 3)...essentially the same as my answer to 1) above. I see "yaru ki" followed by nai/denai most frequently.
"Nanka, yaru ki ga denai." I can't get myself up for doing [something], don't feel like doing something, can't get motivated to [do something].
I have heard "yaru ki (ga) aru" as a positive expression, but not as often as the negative.
"Yaru ki aru ne, anata." You seem to be working hard/pretty excited about it.
"Yoshi, yaru ki ga waite kita" Ok, I'm back on track/ready to do this/pumped up.
And, of course, there's the old standard: "Yaru ki ka?!" (Wanna fight?! Bring it on!)
I would add here that if you're looking to expand your range of expression, look for constructions beyond "yaru ki ga aru/nai" by replacing "yaru" with other Base III (dictionary form) verbs.