I asked my rav this question when my daughter (not bas mitzva yet) frequented a family that didn't "yet" keep a full Shabbos (but they hoped to get there). However, they claimed a strictly kosher home. Based on a perusal of their kitchen, the family integrity and the mother's calls to me with kashrus questions, I believed they kept 100% kosher.
My rav told me that public shmiras shabbos had historically been the basis for kashrus integrity (Shulchan Aruch YD 2:5) because if you did not keep shabbos, you were "gornit"- nothing! Today, we see that many give importance to kashrus over shabbos and lack of keeping shabbos does not undermine their kashrus integrity. My child could eat there.
He then added: "I'm not saying you should eat there, but for your daughter, it's OK."
Summary: It is not best practices to eat by someone with kashrus integrity that doesn't keep shabbos, but you may get a heter from a rav if there are mitigating circumstances.
Your 2nd case is better since there are shomrei shabbos coming in and out (yotzei v'nichnas (see, for example YD 118:10). However, you need to figure out if there is a practical issue besides an halachic one (eg ibid. 12).
Your 3rd case is the worst scenario as he does not have kashrus integrity. The only heter (CYLOR) would be if he is a mumar letayavon- he doesn't keep kosher because he gets a benefit from eating non-kosher, not because he doesn't want to do the mitzva. If the kosher and the non-kosher were equal, he would eat the kosher. So if the kosher burger joint was as close as the non-kosher one and your going to pay him for the exact amount on the receipt, halacha would allow you to trust him (YD 2:2-4), assuming he is either shomer shabbos publicly, or you rely on the aforementioned permit.
I used the above logic to refrain from being mochiach my friend regarding eating by shomrei shabbos who would eat at non-kosher restaurants in a town where kosher was slim-pickings.